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

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)

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)

Absorbing Area Controlling The Global Dynamics Fate of A Locally Repelled Trajectory? Dependent on the existence of an Absorbing Area, acting as a bounded trapping area Attracted to another distant attractor Local Stability Analysis: Complemented by a Study of Global Dynamics

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

Transition from Periodic to Chaotic Synchronization Dissipative Coupling: for =1 Periodic Synchronization Synchronous Chaotic Attractor(SCA)

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)

Riddling Bifurcations All UPOs embedded in the SCA: Transversely Stable (UPOs  Periodic Saddles) Asymptotically (or Strongly) Stable SCA (Lyapunov stable + Attraction in the usual topological sense) Attraction without Bursting for all t 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 SCA Riddling Bifurcation

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

Direct Transition to Global 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 Local to Global Riddling Boundary crisis of an absorbing area Appearance of a new periodic attractor inside the absorbing area

Blow-Out Bifurcations Successive Transverse Bifurcations: Periodic Saddles (PSs)  Periodic Repellers (PRs) (transversely stable) (transversely unstable) {UPOs} = {PSs} + {PRs} Weakly stable SCA (transversely stable): Weight of {PSs} > Weight of {PRs} Blow-out Bifurcation [Weight of {PSs} = Weight of {PRs}] Chaotic Saddle (transversely unstable): Weight of {PSs} < Weight of {PRs}

Global Effect of Blow-out Bifurcations  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

Phase Diagrams for The Chaos Synchronization Unidirectional coupling (=1) Symmetric coupling (=0)

(topological attractor) Summary Investigation of The Mechanism for The Loss of Chaos Synchronization in terms of UPOs embedded in The SCA (Periodic-Orbit Theory) Strongly-stable SCA (topological attractor) Weakly-stable SCA (Milnor attractor) Chaotic Saddle Riddling Bifurcation Blow-out Bifurcation Their Macroscopic Effects depend on The Existence of The Absorbing Area.  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

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

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

Appearance of A Chaotic or Hyperchaotic Attractor through The Blow-out Bifurcations Dissipative Coupling: Hyperchaotic attractor for =0 Chaotic attractor for =1

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) orbit r  0  out-of-phase (asynchronous) orbit PDB (q, r) (2q, r), (2q, r+q)  Symmetric coupling (=0) Conjugate orbit r=q/2: quasi-periodic transition to chaos Other asynchronous orbits: period-doubling transition to chaos

Multistability near The Zero Coupling Critical Point Self-Similar “Topography” of The Parameter Plane Orbits with phase shift q/2 Orbits exhibiting period-doublings Dissipatively-coupled case with for =0

Multistability near The Zero Coupling Critical Point Self-Similar “Topography” of The Parameter Plane Orbits with phase shift q/2 Orbits exhibiting period-doublings Linearly-coupled case with