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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
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
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
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!
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 .....
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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.”
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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.
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
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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 ???
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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)
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
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
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
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
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.
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?
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.
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 !
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.
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
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
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
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
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
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
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 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
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
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