1. 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

2. 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)

3. 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

4. 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

5. 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)

6. 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

7. 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

8. 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

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

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

11. 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

12. 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)

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

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

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