Chaos Synchronization in Coupled Dynamical Systems Sang-Yoon Kim Department of Physics Kangwon National University Synchronization in Coupled Periodic Oscillators Synchronous Pendulum Clocks Synchronously Flashing Fireflies

Synchronization in Coupled Chaotic Oscillators  Lorentz Attractor [ J. Atmos. Sci. 20, 130 (1963)] z Butterfly Effect: Sensitive Dependence on Initial Conditions (small cause  large effect) y x  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)

Private Communication (Application) [K. Cuomo and A. Oppenheim, Phys. Rev. Lett. 71, 65 (1993)] Transmission Using Chaotic Masking (Information signal) Chaotic System Chaotic System  + - Transmitter Receiver

Coupled 1D Maps  1D Map  Coupling function C: coupling parameter  Asymmetry parameter  =0: symmetric coupling  exchange symmetry =1: unidirectional coupling  Invariant Synchronization Line y = x

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)

Transverse Bifurcations of UPOs : Transverse Lyapunov exponent of the SCA (Transverse attraction on the average)  Chaos Synchronization (SCA  Transversely unstable chaotic saddle)  Complete Desynchronization {UPOs} = {Transversely Stable Periodic Saddles (PSs)} + {Transversely Unstable Periodic Repellers (PRs)} “Weight” of {PSs} > (<) “Weight” of {PRs}  Weak Synchronization Strong Synchronization Weak Synchronization C Blowout Bifurcation First Transverse Bifurcation First Transverse Bifurcation Blowout Bifurcation

Global Effect of The First Transverse Bifurcations (Strong Synchronization  Weak Synchronization) e.g. Fate of the Locally Repelled Trajectories? Presence of an absorbing area  Attractor Bubbling Bubbling transition through a supercritical PDB Absence of an absorbing area  Basin Riddling Riddling transition through a transcritical contact bifurcation Basin riddled with a dense set of “holes” leading to divergent orbits Milnor attractor in a measure-theoretical sense

Global Effect of Blow-out Bifurcations (SCA  Chaotic Saddle: Complete Desynchronization)  Absence of an absorbing area (riddled basin) Abrupt Collapse of the Synchronized Chaotic State  Presence of an absorbing area  On-Off Intermittency Appearance of an asynchronous chaotic attractor covering the whole absorbing area

Types of Asynchronous Attractors Born via Blow-out Bifurcations Hyperchaotic attractor for symmetric coupling (=0) Chaotic attractor for unidirectional coupling (=1)

Phase Diagrams for The Chaos Synchronization Dissipatively coupled case with Symmetric coupling (=0) Unidirectional coupling (=1)

Summary Investigation of The Mechanism for The Loss of Chaos Synchronization in terms of Transverse Bifurcations of UPOs embedded in The SCA (Periodic-Orbit Theory) First Transverse Bifurcation Strongly-stable SCA Weakly-stable SCA Chaotic Saddle Blow-out Bifurcation Their Macroscopic Effects depend on The Existence of The Absorbing Area.  Attractor Bubbling  Basin Riddling  Supercritical case  Appearance of An Asynchronous Chaotic Attractor, Exhibiting On-Off Intermittency  Subcritical case  Abrupt Collapse of A Synchronous Chaotic State

Direct Transition to Riddling Symmetric systems Subcritical Pitchfork Bifurcation Contact Bifurcation No Contact (Attractor Bubbling of Hard Type) Asymmetric systems Transcritical Bifurcation Contact Bifurcation No Contact (Attractor Bubbling of Hard Type)

Transition from Bubbling to Riddling Boundary crisis of an absorbing area Appearance of a new periodic attractor inside the absorbing area

Symmetry-Conserving and Breaking Blow-out Bifurcations Linear Coupling: Symmetry-Conserving Blow-out Bifurcation Symmetry-Breaking Blow-out Bifurcation

Phase Diagrams for The Periodic Synchronization Dissipatively coupled case with Symmetric coupling (=0) Unidirectional coupling (=1)