![Page 1: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/1.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
The Topology of Chaos
Robert Gilmore
Physics DepartmentDrexel University
Philadelphia, PA [email protected]
Colloquium, Physics DepartmentUniversity of Georgia, Athens, GA
October 6, 2008
![Page 2: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/2.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
The Topology of Chaos
Robert Gilmore
Physics Department
Drexel University
Philadelphia, PA 19104
Colloquium, Physics DepartmentUniversity of Georgia, Athens, GA
October 9, 2008
![Page 3: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/3.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Table of Contents
Outline1 Overview
2 Experimental Challenge
3 Topology of Orbits
4 Topological Analysis Program
5 Basis Sets of Orbits
6 Bounding Tori
7 Covers and Images
8 Quantizing Chaos
9 Representation Theory of Strange Attractors
10 Summary
![Page 4: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/4.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Background
J. R. Tredicce
Can you explain my data?
I dare you to explain my data!
![Page 5: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/5.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Motivation
Where is Tredicce coming from?
Feigenbaum: α = 4.66920 16091 .....δ = −2.50290 78750 .....
![Page 6: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/6.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Experiment
Laser with Modulated LossesExperimental Arrangement
![Page 7: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/7.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Our Hope
Original Objectives
Construct a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
![Page 8: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/8.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Our Result
Result
There is now a classification theory.
1 It is topological
2 It has a hierarchy of 4 levels
3 Each is discrete
4 There is rigidity and degrees of freedom
5 It is applicable to R3 only — for now
![Page 9: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/9.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Topology Enters the Picture
The 4 Levels of Structure
• Basis Sets of Orbits
• Branched Manifolds
• Bounding Tori
• Extrinsic Embeddings
![Page 10: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/10.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Topological Components
Organization
LINKS OF PERIODIC ORBITSorganize
BOUNDING TORIorganize
BRANCHED MANIFOLDSorganize
LINKS OF PERIODIC ORBITS
![Page 11: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/11.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Experimental Schematic
Laser Experimental Arrangement
![Page 12: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/12.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Experimental Motivation
Oscilloscope Traces
![Page 13: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/13.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Results, Single Experiment
Bifurcation Schematics
![Page 14: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/14.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Some Attractors
Coexisting Basins of Attraction
![Page 15: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/15.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Many Experiments
Bifurcation Perestroikas
![Page 16: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/16.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Real Data
Experimental Data: LSA
Lefranc - Cargese
![Page 17: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/17.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Real Data
Experimental Data: LSA
![Page 18: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/18.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Mechanism
Stretching & Squeezing in a Torus
![Page 19: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/19.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Time Evolution
Rotating the Poincare Sectionaround the axis of the torus
![Page 20: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/20.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Time Evolution
Rotating the Poincare Sectionaround the axis of the torus
Lefranc - Cargese
![Page 21: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/21.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Another Visualization
Cutting Open a Torus
![Page 22: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/22.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Satisfying Boundary Conditions
Global Torsion
![Page 23: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/23.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Experimental Schematic
A Chemical Experiment
The Belousov-Zhabotinskii Reaction
![Page 24: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/24.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Chaos
Chaos
Motion that is
•Deterministic: dxdt = f(x)
•Recurrent
•Non Periodic
• Sensitive to Initial Conditions
![Page 25: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/25.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Strange Attractor
Strange Attractor
The Ω limit set of the flow. There areunstable periodic orbits “in” thestrange attractor. They are
• “Abundant”
•Outline the Strange Attractor
•Are the Skeleton of the StrangeAttractor
![Page 26: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/26.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Skeletons
UPOs Outline Strange attractors
BZ reaction
![Page 27: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/27.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Skeletons
UPOs Outline Strange attractors
Lefranc - Cargese
![Page 28: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/28.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Dynamics and Topology
Organization of UPOs in R3:
Gauss Linking Number
LN(A,B) =1
4π
∮ ∮(rA − rB)·drA×drB
|rA − rB|3
# Interpretations of LN ' # Mathematicians in World
![Page 29: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/29.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Linking Numbers
Linking Number of Two UPOs
Lefranc - Cargese
![Page 30: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/30.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Evolution in Phase Space
One Stretch-&-Squeeze Mechanism
![Page 31: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/31.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Motion of Blobs in Phase Space
Stretching — Squeezing
![Page 32: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/32.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Collapse Along the Stable Manifold
Birman - Williams Projection
Identify x and y if
limt→∞|x(t)− y(t)| → 0
![Page 33: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/33.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
![Page 34: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/34.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
![Page 35: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/35.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
![Page 36: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/36.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Birman-Williams Theorem
Assumptions, B-W Theorem
A flow Φt(x)
• on Rn is dissipative, n = 3, so thatλ1 > 0, λ2 = 0, λ3 < 0.
•Generates a hyperbolic strangeattractor SA
IMPORTANT: The underlined assumptions can be relaxed.
![Page 37: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/37.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Birman-Williams Theorem
Conclusions, B-W Theorem
• The projection maps the strangeattractor SA onto a 2-dimensionalbranched manifold BM and the flow Φt(x)on SA to a semiflow Φ(x)t on BM.•UPOs of Φt(x) on SA are in 1-1correspondence with UPOs of Φ(x)t onBM. Moreover, every link of UPOs of(Φt(x),SA) is isotopic to the correspondlink of UPOs of (Φ(x)t,BM).
Remark: “One of the few theorems useful to experimentalists.”
![Page 38: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/38.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
A Very Common Mechanism
Rossler:
Attractor Branched Manifold
![Page 39: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/39.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
A Mechanism with Symmetry
Lorenz:
Attractor Branched Manifold
![Page 40: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/40.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Examples of Branched Manifolds
Inequivalent Branched Manifolds
![Page 41: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/41.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Aufbau Princip for Branched Manifolds
Any branched manifold can be built upfrom stretching and squeezing units
subject to the conditions:•Outputs to Inputs•No Free Ends
![Page 42: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/42.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Dynamics and Topology
Rossler System
![Page 43: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/43.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Dynamics and Topology
Lorenz System
![Page 44: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/44.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Dynamics and Topology
Poincare Smiles at Us in R3
•Determine organization of UPOs ⇒
•Determine branched manifold ⇒
•Determine equivalence class of SA
![Page 45: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/45.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Topological Analysis Program
Topological Analysis Program
Locate Periodic Orbits
Create an Embedding
Determine Topological Invariants (LN)
Identify a Branched Manifold
Verify the Branched Manifold
—————————————————————————-
Model the Dynamics
Validate the Model
![Page 46: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/46.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Locate UPOs
Method of Close Returns
![Page 47: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/47.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Embeddings
Embeddings
Many Methods: Time Delay, Differential, Hilbert Transforms,SVD, Mixtures, ...
Tests for Embeddings: Geometric, Dynamic, Topological†
None Good
We Demand a 3 Dimensional Embedding
![Page 48: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/48.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Locate UPOs
An Embedding and Periodic Orbits
Lefranc - Cargese
![Page 49: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/49.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Determine Topological Invariants
Linking Number of Orbit Pairs
Lefranc - Cargese
![Page 50: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/50.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Determine Topological Invariants
Compute Table of Expt’l LN
![Page 51: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/51.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Determine Topological Invariants
Compare w. LN From Various BM
![Page 52: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/52.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Determine Topological Invariants
Guess Branched Manifold
Lefranc - Cargese
![Page 53: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/53.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Determine Topological Invariants
Identification & ‘Confirmation’
• BM Identified by LN of small number of orbits
• Table of LN GROSSLY overdetermined
• Predict LN of additional orbits
• Rejection criterion
![Page 54: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/54.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Determine Topological Invariants
What Do We Learn?• BM Depends on Embedding• Some things depend on embedding, some don’t• Depends on Embedding: Global Torsion, Parity, ..• Independent of Embedding: Mechanism
![Page 55: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/55.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Perestroikas of Strange Attractors
Evolution Under Parameter Change
Lefranc - Cargese
![Page 56: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/56.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Perestroikas of Strange Attractors
Evolution Under Parameter Change
Lefranc - Cargese
![Page 57: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/57.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
An Unexpected Benefit
Analysis of Nonstationary Data
Lefranc - Cargese
![Page 58: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/58.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Last Steps
Model the DynamicsA hodgepodge of methods exist: # Methods ' # Physicists
Validate the ModelNeeded: Nonlinear analog of χ2 test. OPPORTUNITY:Tests that depend on entrainment/synchronization.
![Page 59: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/59.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Our Hope → Now a Result
Compare withOriginal Objectives
Construct a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
![Page 60: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/60.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Orbits Can be “Pruned”
There Are Some Missing Orbits
Lorenz Shimizu-Morioka
![Page 61: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/61.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Linking Numbers, Relative Rotation Rates, Braids
Orbit Forcing
![Page 62: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/62.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
An Ongoing Problem
Forcing Diagram - Horseshoe
![Page 63: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/63.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
An Ongoing Problem
Status of Problem
Horseshoe organization - active
More folding - barely begun
Circle forcing - even less known
Higher genus - new ideas required
![Page 64: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/64.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Perestroikas of Branched Manifolds
Constraints on Branched Manifolds
“Inflate” a strange attractor
Union of ε ball around each point
Boundary is surface of bounded 3D manifold
Torus that bounds strange attractor
![Page 65: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/65.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Torus and Genus
Torus, Longitudes, Meridians
![Page 66: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/66.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Flows on Surfaces
Surface Singularities
Flow field: three eigenvalues: +, 0, –
Vector field “perpendicular” to surface
Eigenvalues on surface at fixed point: +, –
All singularities are regular saddles∑s.p.(−1)index = χ(S) = 2− 2g
# fixed points on surface = index = 2g - 2
![Page 67: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/67.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Flows in Vector Fields
Flow Near a Singularity
![Page 68: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/68.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Some Bounding Tori
Torus Bounding Lorenz-like Flows
![Page 69: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/69.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Canonical Forms
Twisting the Lorenz Attractor
![Page 70: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/70.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Constraints Provided by Bounding Tori
Two possible branched manifoldsin the torus with g=4.
![Page 71: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/71.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Use in Physics
Bounding Tori contain all knownStrange Attractors
![Page 72: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/72.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Labeling Bounding Tori
Labeling Bounding Tori
Poincare section is disjoint union of g-1 disks
Transition matrix sum of two g-1 × g-1 matrices
One is cyclic g-1 × g-1 matrix
Other represents union of cycles
Labeling via (permutation) group theory
![Page 73: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/73.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Some Bounding Tori
Bounding Tori of Low Genus
![Page 74: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/74.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Motivation
Some Genus-9 Bounding Tori
![Page 75: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/75.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Aufbau Princip for Bounding Tori
Any bounding torus can be built upfrom equal numbers of stretching andsqueezing units
•Outputs to Inputs•No Free Ends• Colorless
![Page 76: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/76.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Aufbau Princip for Bounding Tori
Application: Lorenz Dynamics, g=3
![Page 77: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/77.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Poincare Section
Construction of Poincare Section
![Page 78: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/78.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Exponential Growth
The Growth is Exponential
![Page 79: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/79.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Exponential Growth
The Growth is ExponentialThe Entropy is log 3
![Page 80: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/80.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Extrinsic Embedding of Bounding Tori
Extrinsic Embedding of Intrinsic Tori
Partial classification by links of homotopy group generators.Nightmare Numbers are Expected.
![Page 81: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/81.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Modding Out a Rotation Symmetry
Modding Out a Rotation Symmetry X
YZ
→ u
vw
=
Re (X + iY )2
Im (X + iY )2
Z
![Page 82: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/82.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Lorenz Attractor and Its Image
![Page 83: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/83.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Lifting an Attractor: Cover-Image Relations
Creating a Cover with Symmetry X
YZ
← u
vw
=
Re (X + iY )2
Im (X + iY )2
Z
![Page 84: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/84.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Cover-Image Related Branched Manifolds
Cover-Image Branched Manifolds
![Page 85: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/85.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Covering Branched Manifolds
Two Two-fold LiftsDifferent Symmetry
Rotation InversionSymmetry Symmetry
![Page 86: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/86.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Topological Indices
Topological Index: Choose Group
Choose Rotation Axis (Singular Set)
![Page 87: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/87.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Locate the Singular Set wrt Image
Different Rotation Axes ProduceDifferent (Nonisotopic) Lifts
![Page 88: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/88.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Nonisotopic Locally Diffeomorphic Lifts
![Page 89: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/89.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Indices (0,1) and (1,1)
Two Two-fold CoversSame Symmetry
![Page 90: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/90.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Indices (0,1) and (1,1)
Three-fold, Four-fold Covers
![Page 91: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/91.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Two Inequivalent Lifts with V4 Symmetry
![Page 92: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/92.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
How to Construct Covers/Images
Algorithm
• Construct Invariant Polynomials, Syzygies, Radicals
• Construct Singular Sets
• Determine Topological Indices
• Construct Spectrum of Structurally Stable Covers
• Structurally Unstable Covers Interpolate
![Page 93: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/93.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Surprising New Findings
Symmetries Due to Symmetry
Schur’s Lemmas & Equivariant Dynamics
Cauchy Riemann Symmetries
Clebsch-Gordon Symmetries
Continuations
Analytic ContinuationTopological ContinuationGroup Continuation
![Page 94: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/94.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Covers of a Trefoil Torus
Granny Knot Square Knot
Trefoil Knot
![Page 95: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/95.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
You Can Cover a Cover = Lift a Lift
Covers of Covers of Covers
Rossler Lorenz
Ghrist
![Page 96: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/96.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Universal Branched Manifold
EveryKnot Lives Here
![Page 97: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/97.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Isomorphisms and Diffeomorphisms
Local Stuff
Groups:Local IsomorphismsCartan’s Theorem
Dynamical Systems:Local Diffeomorphisms??? Anything Useful ???
![Page 98: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/98.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Universal Covering Group
Cartan’s Theorem for Lie Groups
![Page 99: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/99.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Universal Image Dynamical System
Locally Diffeomorphic Covers of D
D: Universal Image Dynamical System
![Page 100: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/100.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 101: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/101.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 102: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/102.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 103: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/103.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 104: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/104.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 105: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/105.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 106: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/106.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 107: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/107.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Creating New Attractors
Rotating the Attractor
d
dt
[XY
]=[F1(X,Y )F2(X,Y )
]+[a1 sin(ωdt+ φ1)a2 sin(ωdt+ φ2)
][u(t)v(t)
]=[
cos Ωt − sin Ωtsin Ωt cos Ωt
] [X(t)Y (t)
]d
dt
[uv
]= RF(R−1u) +Rt + Ω
[−v+u
]Ω = n ωd q Ω = p ωd
Global Diffeomorphisms Local Diffeomorphisms(p-fold covers)
![Page 108: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/108.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Two Phase Spaces: R3 and D2 × S1
Rossler Attractor: Two Representations
R3 D2 × S1
![Page 109: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/109.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Other Diffeomorphic Attractors
Rossler Attractor:
Two More Representations with n = ±1
![Page 110: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/110.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Subharmonic, Locally Diffeomorphic Attractors
Rossler Attractor:
Two Two-Fold Covers with p/q = ±1/2
![Page 111: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/111.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Subharmonic, Locally Diffeomorphic Attractors
Rossler Attractor:
Two Three-Fold Covers with p/q = −2/3,−1/3
![Page 112: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/112.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Subharmonic, Locally Diffeomorphic Attractors
Rossler Attractor:
And Even More Covers (with p/q = +1/3,+2/3)
![Page 113: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/113.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
New Measures
Angular Momentum and Energy
L(0) = limτ→∞
1τ
∫ τ
0XdY−Y dX
L(Ω) = 〈uv − vu〉
= L(0) + Ω〈R2〉
K(0) = limτ→∞
1τ
∫ τ
0
12
(X2+Y 2)dt
K(Ω) = 〈12
(u2 + v2)〉
= K(0) + ΩL(0) +12
Ω2〈R2〉
〈R2〉 = limτ→∞
1τ
∫ τ
0(X2 + Y 2)dt = lim
τ→∞
1τ
∫ τ
0(u2 + v2)dt
![Page 114: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/114.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
New Measures, Diffeomorphic Attractors
Energy and Angular Momentum
Diffeomorphic, Quantum Number n
![Page 115: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/115.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
New Measures, Subharmonic Covering Attractors
Energy and Angular Momentum
Subharmonics, Quantum Numbers p/q
![Page 116: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/116.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Embeddings
Embeddings
An embedding creates a diffeomorphism between an(‘invisible’) dynamics in someone’s laboratory and a (‘visible’)attractor in somebody’s computer.
Embeddings provide a representation of an attractor.
Equivalence is by Isotopy.
Irreducible is by Dimension
![Page 117: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/117.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Representation Labels
Inequivalent Irreducible Representations
Irreducible Representations of 3-dimensional Genus-oneattractors are distinguished by three topological labels:
ParityGlobal TorsionKnot Type
PNKT
ΓP,N,KT (SA)
Mechanism (stretch & fold, stretch & roll) is an invariant ofembedding. It is independent of the representation labels.
![Page 118: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/118.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Creating Isotopies
Equivalent Reducible Representations
Topological indices (P,N,KT) are obstructions to isotopy forembeddings of minimum dimension (irreduciblerepresentations).
Are these obstructions removed by injections into higherdimensions (reducible representations)?
Systematically?
![Page 119: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/119.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Creating Isotopies
Equivalences by InjectionObstructions to Isotopy
R3
Global TorsionParityKnot Type
→ R4
Global Torsion
→ R5
There is one Universal reducible representation in RN , N ≥ 5.In RN the only topological invariant is mechanism.
![Page 120: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/120.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
The Road Ahead
Summary
1 Question Answered ⇒
2 Questions Raised
We must be on the right track !
![Page 121: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/121.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Our Hope
Original Objectives Achieved
There is now a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
![Page 122: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/122.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Our Result
Result
There is now a classification theory
for low-dimensional strange attractors.
1 It is topological
2 It has a hierarchy of 4 levels
3 Each is discrete
4 There is rigidity and degrees of freedom
5 It is applicable to R3 only — for now
![Page 123: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/123.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 124: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/124.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 125: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/125.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 126: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/126.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 127: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/127.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 128: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/128.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Four Levels of Structure
![Page 129: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/129.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Topological Components
Poetic Organization
LINKS OF PERIODIC ORBITSorganize
BOUNDING TORIorganize
BRANCHED MANIFOLDSorganize
LINKS OF PERIODIC ORBITS
![Page 130: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/130.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Answered Questions
Some Unexpected ResultsPerestroikas of orbits constrained by branched manifoldsRoutes to Chaos = Paths through orbit forcing diagramPerestroikas of branched manifolds constrained bybounding toriGlobal Poincare section = union of g − 1 disksSystematic methods for cover - image relationsExistence of topological indices (cover/image)Universal image dynamical systemsNLD version of Cartan’s Theorem for Lie GroupsTopological Continuation – Group ContinuuationCauchy-Riemann symmetriesQuantizing ChaosRepresentation labels for inequivalent embeddingsRepresentation Theory for Strange Attractors
![Page 131: The Topology of Chaos - Physics Departmentbob/Presentations/univ_georgia.pdf · The Topology of Chaos Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 robert.gilmore@drexel.edu](https://reader034.vdocuments.mx/reader034/viewer/2022050407/5f847419a8e64648f328a932/html5/thumbnails/131.jpg)
The Topologyof Chaos
RobertGilmore
Introduction-01
Introduction-02
Overview-01
Overview-02
Overview-03
Overview-04
Overview-05
Overview-06
Overview-07
Experimental-01
Experimental-02
Experimental-03
Experimental-04
Experimental-05
Experimental-06a
Experimental-06b
Experimental-07
Experimental-08a
Experimental-08b
Experimental-09
Experimental-10
Experimental-11
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Program-01
Program-02
Program-03
Program-04
Program-05
Program-06
Program-07
Program-08
Program-09
Program-10
Program-11a
Program-11b
Program-12
Program-13
Program-14
Basis Sets ofOrbits-01
Basis Sets ofOrbits-02
Basis Sets ofOrbits-03
Basis Sets ofOrbits-04
BoundingTori-01
BoundingTori-02
BoundingTori-03
BoundingTori-04
BoundingTori-05
BoundingTori-06
BoundingTori-07
BoundingTori-08
BoundingTori-09
BoundingTori-10
BoundingTori-11
BoundingTori-12
BoundingTori-13
BoundingTori-14
BoundingTori-15
BoundingTori-16
BoundingTori-17
Covers-01
Covers-02
Covers-03
Covers-04
Covers-05
Covers-06
Covers-07
Covers-08
Covers-09
Covers-10
Covers-11
Covers-12
Covers-13
Covers-14
Covers-15a
Covers-15b
Covers-16
Covers-17
Covers-18
Covers-19
QuantizingChaos-01
QuantizingChaos-02
QuantizingChaos-03
QuantizingChaos-04
QuantizingChaos-05
QuantizingChaos-06
QuantizingChaos-07
QuantizingChaos-08
QuantizingChaos-09
RepresentationTheory-01
RepresentationTheory-02
RepresentationTheory-03
RepresentationTheory-04
Summary-01
Summary-02
Summary-03
Summary-04
Summary-05
Summary-06
Summary-07
Summary-08
Unanswered Questions
We hope to find:Robust topological invariants for RN , N > 3A Birman-Williams type theorem for higher dimensions
An algorithm for irreducible embeddings
Embeddings: better methods and tests
Analog of χ2 test for NLD
Better forcing results: Smale horseshoe, D2 → D2,n×D2 → n×D2 (e.g., Lorenz), DN → DN , N > 2Representation theory: complete
Singularity Theory: Branched manifolds, splitting points(0 dim.), branch lines (1 dim).
Singularities as obstructions to isotopy