1

Synchronization in Coupled Chaotic Systems

Sang-Yoon Kim

Department of Physics

Kangwon National University

Korea

Synchronization in Coupled Periodic Oscillators

Synchronous Pendulum Clocks Synchronously Flashing Fireflies

2

Chaos and Synchronization

Lorenz Attractor [ Lorenz, J. Atmos. Sci. 20, 130 (1963)]

Coupled Brusselator Model (Chemical Oscillators)

H. Fujisaka and T. Yamada, “Stability Theory of Synchronized Motion in Coupled-Oscillator Systems,” Prog. Theor. Phys. 69, 32 (1983)

z

yx

Butterfly Effect: Sensitive Dependence on Initial Conditions (small cause large effect)

3

ChaoticSystem + Chaotic

System -

ts

ty ty ts

Secure Communication (Application)

Transmission Using Chaotic Masking

Transmitter Receiver

(Secret Message)

Several Types of Chaos Synchronization

Different degrees of correlation between the interacting subsystems

Identical Subsystems Complete Synchronization Nonidentical Subsystems Generalized Synchronization Phase Synchronization Lag Synchronization

Secret Message Spectrum

Chaotic MaskingSpectrum

Frequency (kHz)

4

21 1)( ttt Axxfx

An infinite sequence of period doubling bifurcations ends at a finite accumulation point 506092189155401.1A

When exceed , a chaotic attractor with positive Lyapunov exponent appears.

A

1210 tt xxxxxIterates: (trajectory) Attractor

tedtd )0()(

A

(x: seasonly breeding inset population)

1D Map (Building Blocks)

Complete Synchronization in Coupled Chaotic 1D Maps

Period-Doubling Transition to Chaos

5

).,()(

),,(1)(:

1

1

tttt

tttt

xygCyfy

yxgCxfxT

Coupling function

...,2,1)(),()(, nxxuxuyuyxg n

C: coupling parameter

Asymmetry parameter = 0: symmetric coupling exchange symmetry = 1: unidirectional coupling

Invariant synchronization line y = x

Synchronous orbits on the diagonal Asynchronous orbits off the diagonal

1.0,1 CA

22, xyyxg

Coupled 1D Maps

1

6

Transverse Stability of The Synchronous Chaotic Attractor

Synchronous Chaotic Attractor (SCA) on The Invariant Synchronization Line

SCA: Stable against the “Transverse Perturbation” Chaos Synchronization

An infinite number of Unstable Periodic Orbits (UPOs) embedded in the SCA and forming its skeleton Characterization of the Macroscopic Phenomena Associated with the Transverse Stability of the SCA in terms of UPOs (Periodic-Orbit Theory)

7

WeakSynchronization

WeakSynchronization

StrongSynchronization

Transverse Bifurcations of UPOs

: Transverse Lyapunov exponent of the SCA (determining local transverse stability) 0 (SCA Transversely stable) Chaos Synchronization

(SCA Transversely unstable chaotic saddle) Complete Desynchronization

0

{UPOs} = {Transversely Stable Periodic Saddles (PSs)} + {Transversely Unstable Periodic Repellers (PRs)}

BlowoutBifurcation

FirstTransverseBifurcation

FirstTransverseBifurcation

BlowoutBifurcation

“Weight” of {PSs} > (<) “Weight” of {PRs} 00

0

0 0 0

0

C

Investigation of transverse stability of the SCA in terms of UPOs

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Strong Synchronization

All UPOs embedded in the SCA: Transversely stable

SCA: Asymptotically stable(Lyapunov stable + Attraction in the usual topological sense)

Attraction without bursting for all t

2,82.1 CA

e.g. Unidirectionally and Dissipatively Coupled Case with = 1 and g(x, y) = y2x2

Strong synchronization for A = 1.82 and Ct,l (= 2.789 …) < C < Ct,r (= 0.850 …)

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Global Effect of The First Transverse Bifurcation

Fate of A Locally Repelled Trajectory?

Attracted to another distant attractor

Transverse Bifurcation through which a first periodic saddle becomes transversely unstable

Local Bursting

Lyapunov unstable(Loss of Asymptotic

Stability)

Dependent on the existence of an Absorbing Area, controlling the global dynamics and acting as a bounded trapping area

Folding backof repelled trajectory

Local Stability Analysis: Complemented by a Study of Global Dynamics

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Bubbling Transition through The 1st Transverse Bifurcation

C

Strong synchronization BubblingRiddling

...789.2, ltC ...850.0, rtC

Case of rtCC ,Presence of an absorbing area Bubbling Transition

Noise and Parameter Mismatching Persistent intermittent bursting (Attractor Bubbling)

Transient intermittent bursting

Transcritical Contact Bif. Supercritical Period-Doubling Bif.

68.0,82.1 CA

005.0,68.0,82.1 CA

68.0,82.1 CA

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C

Strong synchronization BubblingRiddling

...789.2, ltC ...850.0, rtC

Case of ltCC ,

Transcritical Contact Bif. Supercritical Period-Doubling Bif.

Riddling Transition through The 1st Transverse Bifurcation

Disappearance of an absorbing area Riddling Transition

ltCC ,

an absorbing area surrounding the SCA

Contact between the SCA andthe basin boundary

ltCC ,67.2C

ltCC ,

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Riddled Basin

After the transcritical contact bifurcation, the basin becomes “riddled” with a dense set of “holes” leading to divergent orbits. The SCA is no longer a topological attractor; it becomes a Milnor attractor in a measure-theoretical sense.

As C decreases from Ct,l, the measure of the riddled basin decreases.

88.2C 93.2C

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Characterization of The Riddled Basin

Divergence Exponent

Divergence probability P(d) ~ d

(Take many randomly chosen initial conditions on the line y=x+d and determine which basin they lie in.) Measure of the Basin Riddling

Uncertainty Exponent

Uncertainty probability P() ~

(Take two initial conditions within a small square with sides of length 2 inside the basin and determine the final states of the trajectories starting with them.) Fine Scaled Riddling of the SCA

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Direct Transition to Bubbling or Riddling

Asymmetric systemsTranscritical bifurcation

Subcritical pitchfork or period-doubling bifurcation

Contact bifurcation(Riddling)

Non-contact bifurcation(Bubbling of hard type)

Symmetric systems(Supercritical bifurcations Bubbling transition of soft type)

Contact bifurcation(Riddling)

Non-contact bifurcation(Bubbling of hard type)

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Transition from Bubbling to Riddling

Boundary crisis of an absorbing area

Appearance of a new periodic attractor inside the absorbing area

Bubbling Riddling

Bubbling Riddling

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Basin Riddling through A Dynamic Stabilization

34.0,65.1 CA 26.0,65.1 CA

Symmetrically and dissipatively coupled case with =0 and 22, xyyxg

Bubbling Riddling

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Global Effect of Blow-out Bifurcations

C

Strong synchronization

rtC ,

BubblingRiddling

999.2, lbC 677.0, rbCltC ,

65.0,82.1 CA 65.0,82.1 CA

Weight of {PRs} > Weight of {PSs} SCA Transversely Unstable Chaotic Saddle Complete Desynchronization

Successive Transverse Bifurcations: Periodic Saddles (PSs) Periodic Repellers (PRs) (transversely stable) (transversely unstable)

For C < Cb,l, absence of an absorbing area Subcritical blow-out bifurcation Abrupt collapse of the synchronized chaotic state

For C > Cb,r, presence of an absorbing area Supercritical blow-out bifurcation Appearance of an asynchronous chaotic attractor covering the whole absorbing area and exhibiting the On-Off Intermittency

~ ~Blow-out Bif. First Transverse Bif. First Transverse Bif. Blow-out Bif.

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Symmetry-Conserving and -Breaking Blow-out Bifurcations Symmetrically and linearly coupled case with =0 and

Depending on the shape of a minimal invariant absorbing area, symmetry may or may not be conserved.

028.1,44.1

CA

xyyxg ),(

024.1,44.1

CA

031.1,427.1

CA

027.1,427.1

CA

Symmetry-Conserving Blow-out Bifurcation Symmetry-Breaking Blow-out Bifurcation

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Type of Asynchronous Attractors Born via Blow-out Bif.

{Asynchronous UPOs inside an absorbing area}={Asynchronous PSs with one unstable direction} +{Asynchronous PRs with two unstable directions}

Hyperchaotic attractor for =0 Chaotic attractor for =1

Weight of {PRs} > Weight of {PSs} Weight of {PRs} < Weight of {PSs}

333.0

,83.1

C

A

681.0

,83.1

C

A

Numbers of the period-11 saddles (Ns) and repellers (Nr):

Nr > Ns for < 0.8 Nr < Ns for > 0.9

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Phase Diagram for The Chaos Synchronization

Symmetric coupling (=0)

22, xyyxg Dissipatively coupled case with

Hatched Region: Strong Synchronization, Light Gray Region: Bubbling, Dark Gray Region: RiddlingSolid or Dashed Lines: First Transverse Bifurcation Lines, Solid Circles: Blow-out Bifurcation

Unidirectional coupling (=1)

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First TransverseBifurcation

Their Macroscopic Effects depend on The Existence of The Absorbing Area.

Blow-out Bifurcation

Investigation of The Mechanism for The Loss of Chaos Synchronization in terms of Transverse Bifurcations of UPOs embedded in The SCA (Periodic-Orbit Theory)

Attractor Bubbling

Basin Riddling

Subcritical case Abrupt Collapse of A Synchronous Chaotic State

Supercritical case Appearance of An Asynchronous Chaotic Attractor, Exhibiting The On-Off Intermittency.

ChaoticSaddle

Weakly-stableSCA

Strongly-stableSCA

Summary

References

[1] S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001). [2] S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 105, 187-196 (2001). [3] S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001).

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First TransverseBifurcation

Their Macroscopic Effects depend on The Existence of The Absorbing Area.

Blow-out Bifurcation

Investigation of The Mechanism for The Loss of Chaos Synchronization in terms of Transverse Bifurcations of UPOs embedded in The SCA (Periodic-Orbit Theory)

Attractor Bubbling

Basin Riddling

Subcritical case Abrupt Collapse of A Synchronous Chaotic State

Supercritical case Appearance of An Asynchronous Chaotic Attractor. The type (Symmetric or Asymmetric, Chaotic or Hyperchaotic) of which is determined by an absorbing area.

ChaoticSaddle

Weakly-stableSCA

Strongly-stableSCA

Summary

References

[1] S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001). [2] S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 105, 187-196 (2001). [3] S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001).

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Universality for The Chaos Synchronization

Mechanisms for The Loss of Chaos Synchronization in Coupled 1D Maps Are these mechanisms still valid for the real systems such as the coupled Hénon maps and coupled oscillators?

I think that those mechanisms are Universal ones, independently of the details of coupled systems, based on our preliminary results.

Universality for The Periodic Synchronization (well understood) The coupled 1D maps and coupled oscillators have the phase diagrams of the same structure and they exhibit the same scaling behavior on their critical set.

I believe that there may exist some kind of Universality for both the Chaotic and Periodic Synchronization in Coupled Dynamical Systems.

I suggest the Experimentalists to confirm this kind of universality in real experiment such as the electronic-circuit experiment.

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Phase Diagram for The Chaos Synchronization

Unidirectional coupling (=1)Symmetric coupling (=0)

xyyxg ,Linearly coupled case with

Hatched Region: Strong Synchronization, Light Gray Region: Bubbling, Dark Gray Region: RiddlingSolid or Dashed Lines: First Transverse Bifurcation Lines, Solid Circles: Blow-out BifurcationOpen Circles: Bdry. Crisis of An Absorbing Area, Open Squares: Bdry. Crisis of An Asyn. Chaotic Attractor

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Destruction of Hyperchaotic Attractorsthrough The Dynamic Stabilization

When a dynamic stabilization occurs before the blow-out bifurcation,a transition from bubbling to riddling takes place.

However, a sudden destruction of a hyperchaotic attractor occurswhen such a dynamic stabilization occurs after a blow-out bifurcation.

35.0,84.1 CA 328.0,84.1 CA

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Phase Diagram for Destruction of Hyperchaotic Attractors

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Phase Diagram for The Periodic Synchronization

Unidirectional coupling (=1)Symmetric coupling (=0)

22, xyyxg Dissipatively coupled case with

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Phase Diagram for The Periodic Synchronization

Unidirectional coupling (=1)Symmetric coupling (=0)

xyyxg ,Linearly coupled case with

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Effect of Parameter Mismatch and Noise for The Bubbling Case

Parameter mismatch or noise The SCA is broken up, and then it exhibits a persistent intermittent bursting. Attractor bubbling

01.0,75.0,82.1 CA

005.001.0

The maximum bursting amplitude increases when passing C=Ct,r.

bcbc

|y-x| max

|y-x| max

Ct,r Ct,r

005.0,75.0,82.1 CA

.1

,1:

2221

21

ttttt

tt

yxCyAy

AxxT

: Mismatching parameter

: Noise strength

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Abrupt Change of The Maximum Bursting AmplitudeThe maximum bursting amplitude increases abruptly through the interior crisis of the absorbing area for C 0.8437

Small absorbing area before the crisis Large absorbing area after the crisis

8438.0C 8.0C

Abrupt increase of the maximum bursting amplitude is in contrast to the case of symmetric coupling.

01.0Symmetric coupling (=0)

01.0

bc

|y-x| max

Unidirectional coupling (=1)|y-x| max

~

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Effect of Parameter Mismatch and Noise for The Riddling Case

0 ,91.2 ,82.1 CA 005.0

SCA with the riddled basin Chaotic transientParameter mismatch or noise

001.0

32

8.2C9228.1

: Average life-time of the chaotic transient

Exponential scaling(long lived chaotic transient)

Algebraic scalingC

~2/1

~ e

Ct,lC 2.84

Crossover

Characterization of The Chaotic Transients

8.2C9228.1

~2/1

~ e

~