Markov Chain Modeling and Analysisof Complicated Phenomena inCoupled Chaotic Oscillators

Yoshifumi NISHIO

Dept. of Electrical and Electronic EngineeringTokushima University, JAPAN

Contents

1. Introduction2. Synchronization of Oscillators3. Markov Chain Modeling and Analysis4. Conclusions

Oscillation (Rhythm) Most important signal in natural and

artificial systems

Synchronization Basic phenomenon in higher-

dimensional nonlinear systems

1. Introduction

Analysis is important for ☆ Better understanding of natural phenomena ☆ Future engineering applications

☆ Modeling and Control of ComplexNetworks in Natural and SocialFields

☆ Information Processing with PhasePatterns of Pulse Trains in Brain

☆ Synchronization and Control ofCommunication Networks

☆ Gait Patterns of Walking Robot

Possible Applications

2. Synchronization of Oscillators

Coupling of two or more oscillators with similar natural frequencies

Mutual synchronization( with a constant phase difference )

2a. van der Pol Oscillator

van der Pol oscillator v-i characteristics ofnonlinear resistor

Circuit Equations

Oscillation Waveform

Computer simulatedresults

Circuit experimental results

2b. Resistance Coupling

Circuit R1

Circuit R2

Circuit R1

Two oscillators aresynchronized with 0degree phase difference.

( In-phase synchronization)

Circuit R2

Two oscillators aresynchronized with 180degree phase difference.

( Anti-phasesynchronization )

2c. Reactance Coupling

Two synchronization modes ( in-phase and anti-phase ) coexist.

2d. Large Scale Networks

Generation ofvarious spatial patterns

Propagation oflocal synchronization states

Star Coupled Oscillators

N van der Pol oscillators arecoupled by one resistor.

N=5 (5-phase sync.)

2e. Effect of Frustration

Simple synchronization

(In-phase, Anti-Phase)

Coexistense of complicated phase patterns

☆ Complicated behaviorcaused by some instability

☆ Future engineering applicationexploiting the complicated behavior

3. Markov Chain Modeling and AnalysisIf van der Pol oscillators in a coupled system

are replaced by chaotic oscillators, … ?

Periodic Oscillator Networks

Coexistense of periodic patterns

Quasi-synchronization (desynchronization)

Chaotic Oscillator Networks

Switching of phase states caused byinstability of chaos

Possible applications

Theoretical analysis is difficult.

We have to develop several toolsto reveal the essence of thecomplicated phenomena.

3a. Coupled Chaotic Oscillators

Four chaotic oscillators coupled by one resistor

Chaotic attractor

Chaos is non-periodic, but attractor has a structure.

x I

vz

Computer simulatedresults

Circuit experimental results

)4,3,2,1( =k

nonlinear function

)|1|1(5.0)( +!+= kkk yyyf ""

Circuit equations

kkk

kkkkk

j

jkkkk

yxd

dz

yfzyxd

dy

xzyxd

dx

+=

!!+=

!!+= "=

#

$%#

&$#

)}()({

)(4

1

Coupling resistor

Four-phase sync of chaos

6 phasestates coexist

)2,

2

3,,0(),

2

3,2,,0(),

2,

2

3,,0(

),2

3,2,,0(),,

2

3,2,0(),

2

3,,

2,0(:),,,( 4321

!!!

!!!

!!!

!!!!

!!!!

!xxxx

Computer simulatedresults

Circuit experimental results

Poincare maps

0,011<= xz

Circuit 1 Circuit２ Circuit３ Ciscuit４

Poincare Section

1x

1z

2z 3

z4z

2x

3x

4x

Time series of Poincare map

4-phase sync of chaos Desyncronization

)500(30.0 !"= R# )780(46.0 !"= R#

Dependent angle variable

))(),(( 22 nznx)(1 n!

)(1 n! Phase of circuit 2

Reference: circuit 1

)(2 n! Phase of circuit 3

)(3 n! Phase of circuit 4

Switching of sync. patterns

Time series of

)500(30.0 !"= R# )780(46.0 !"= R#

)(nk

!

Chaotic (unpredictable) Next switching Next phase pattern

4-phase sync of chaos

3b. Statistical analysis using angle

variables

Switching frequencyAverage sojourn timeetc.　　　could be clarified.

Definition of all phase patterns

Markov chain modeling

Six basic synchronized phase patterns

Three intermediate phase patterns

Better understanding

of large scale chaotic networks

Engineering application

of chaotic switchings

654321,,,,, SSSSSS

321,,

IIISSS

State-transition diagram

1SOnly transitions from 1I

SOnly transitions from

Transition probability matrix

The behavior of the Markov chain model canbe described by this transition probabilitymatrix.

Basic quantities

Stationaryprobability

Probability densityfunction of sojourntimeExpectedsojourn time

Second-order Markov chain

More detailed modeling

Transition probability matrix is 57× 57 .

Due to the transition conditions, the number of non-zero elements is 369 out of 57× 57 = 3249.

By virtue of the symmetry of the coupling structureof the original circuit, the number of necessarytransition probabilities is 52.

Simulated results 1

Stationary probabilityand expected sojourn time

Simulated results 2

Probability density function of sojourn time

Six basic synchronized phasepatterns

Three intermediate states

3c. Inductively Coupled Chaotic

Oscillators

Clustering phenomenon

N=6

State-transition diagram

Only transitions from1S

Switching of cluster types

Computer simulation Markovchain

4. Conclusions.

Coupled oscillatory circuits

Coexistence of synchronization states

　　Interesting phenomena　　Various patterns

Statistical analysis of chaotic switching

　　Angle variable and definition of states　　Markov chain modeling