1. 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

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. 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)

4. 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

5. 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

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

7. 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)

8. 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

9. 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

10. 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)

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

12. 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}

13. 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

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

15. (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

16. 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

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

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

19. 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

20. 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

21. 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