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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