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Periodic Orbit Theory for The Chaos Synchronization

Sang-Yoon Kim

Department of Physics

Kangwon National University

Synchronization in Coupled Periodic Oscillators

Synchronous Pendulum Clocks Synchronously Flashing Fireflies

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Synchronization in Coupled Chaotic Oscillators

Lorentz Attractor [ 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)

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

z

yx

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Transverse Stability of The Synchronous Chaotic Attractor

Synchronous Chaotic Attractor (SCA) on The Invariant Synchronization Line in The x-y State Space

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)

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Absorbing Area Controlling The Global Dynamics

Dependent on the existence of an Absorbing Area, acting as a bounded trapping area

Fate of A Locally Repelled Trajectory?

Local Stability Analysis: Complemented by a Study of Global Dynamics

Attracted to another distant attractor

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Coupled 1D Maps

).,()(

),,(1)(:

1

1

tttt

tttt

xygCyfy

yxgCxfxT

1D Map

21 1 ttt Axxfx

Coupling function

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

C: coupling parameter

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

Invariant Synchronization Line y = x

1,1.0,1 CA

22, xyyxg

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Dissipative Coupling: for =1

Transition from Periodic to Chaotic Synchronization

22),( xyyxg

Periodic Synchronization Synchronous Chaotic Attractor(SCA)

2,82.1 CA...155401.1A

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Phase Diagram for The Chaos Synchronization

Strongly stable SCA (hatched region)

Riddling Bifurcation

Weakly stable SCA with locally riddled basin (gray region) with globally riddled basin (dark gray region)

Blow-out Bifurcation

Chaotic Saddle (white region)

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All UPOs embedded in the SCA: Transversely Stable (UPOs Periodic Saddles)

Asymptotically (or Strongly) Stable SCA (Lyapunov stable + Attraction in the usual topological sense)

e.g.

A First Transverse Bifurcation through which a periodic saddle becomes transversely unstable

Local Bursting

Lyapunov unstable(Loss of Asymptotic Stability)

Strongly stable SCA Weakly stable SCARiddlingBifurcation

850.0789.2,82.1 ,, rrlr CCCA

Riddling Bifurcations

Attraction without Burstingfor all t

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Global Effect of The Riddling Bifurcations

Fate of the Locally Repelled Trajectories?

Presence of an absorbing area Attractor Bubbling Local Riddling Transition through A Supercritical PDB

Absence of an absorbing area Riddled Basin Global Riddling Transition through A Transcritical Contact Bifurcation

68.0,82.1

CA 005.0,68.0,82.1 CA

67.2,82.1 CA 93.2,82.1 CAlrCCA ,,82.1

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Direct Transition to Global Riddling Symmetric systems

Asymmetric systemsTranscritical Bifurcation

Subcritical Pitchfork Bifurcation

Contact Bifurcation

No Contact(Attractor Bubbling ofHard Type)

Contact Bifurcation

No Contact(Attractor Bubbling ofHard Type)

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Transition from Local to Global Riddling Boundary crisis of an absorbing area

Appearance of a new periodic attractor inside the absorbing area

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Blow-Out Bifurcations

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

{UPOs} = {PSs} + {PRs}

Chaotic Saddle (transversely unstable): Weight of {PSs} < Weight of {PRs}

Blow-out Bifurcation[Weight of {PSs} = Weight of {PRs}]

Weakly stable SCA (transversely stable): Weight of {PSs} > Weight of {PRs}

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Global Effect of Blow-out Bifurcations

65.0,82.1 CA 65.0,82.1 CA

Absence of an absorbing area (globally riddled basin) Abrupt Collapse of the Synchronized Chaotic State

Presence of an absorbing area (locally riddled basin) On-Off Intermittency Appearance of an asynchronous chaotic attractor covering the whole absorbing area

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Phase Diagrams for The Chaos Synchronization

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

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Summary

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

Blow-out Bifurcation

Strongly-stableSCA

(topological attractor)

Weakly-stableSCA

(Milnor attractor)

ChaoticSaddle

RiddlingBifurcation

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

Local riddling Attractor Bubbling

Global riddling Riddled Basin of Attraction

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

Subcritical case Abrupt Collapse of A Synchronous Chaotic State

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ChaoticSystem + Chaotic

System -

ts

ty ty ts

Private Communication (Application)[K. Cuomo and A. Oppenheim, Phys. Rev. Lett. 71, 65 (1993)]

Transmission Using Chaotic Masking

Transmitter Receiver

(Information signal)

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Symmetry-Conserving and Breaking Blow-out Bifurcations

028.1,44.1

CA

xyyxg ),(Linear Coupling:

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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Appearance of A Chaotic or Hyperchaotic Attractor through The Blow-out Bifurcations

Hyperchaotic attractor for =0

bCCCCA 008.0,83.1 Chaotic attractor for =1

22),( xyyxg Dissipative Coupling:

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Classification of Periodic Orbits in Coupled 1D Maps

For C=0, periodic orbits can be classified in terms of period q and phase shift r

For each subsystem, attractor

The composite system has different attractors distinguished by a phase shift r,

r = 0 in-phase (synchronous) orbitr 0 out-of-phase (asynchronous) orbit

(q, r) (2q, r), (2q, r+q)PDB

nq 2

n2 rtt yx

Symmetric coupling (=0)

,12,...,1

2,12,...,1,0

qq

qqr

r=q/2: quasi-periodic transition to chaosOther asynchronous orbits: period-doubling transition to chaos

Conjugate orbit

0,27.1 CA

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Multistability near The Zero Coupling Critical Point

Orbits with phase shift q/2 Orbits exhibiting period-doublings

Dissipatively-coupled case with for =022),( xyyxg

Self-Similar “Topography” of The Parameter Plane

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Multistability near The Zero Coupling Critical Point

Orbits with phase shift q/2 Orbits exhibiting period-doublings

Linearly-coupled case with xyyxg ),(

Self-Similar “Topography” of The Parameter Plane