Periodic%20Orbit%20Theory%20for%20The%20Chaos%20Synchronization

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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)
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)

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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
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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
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Transition from Periodic to Chaotic
Synchronization
Dissipative Coupling
for ?1
Synchronous Chaotic Attractor(SCA)
Periodic Synchronization
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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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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
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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

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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)
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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
Weakly stable SCA (transversely stable) Weight
of PSs gt Weight of PRs
Blow-out Bifurcation Weight of PSs Weight of
PRs
Chaotic Saddle (transversely unstable) Weight of
PSs lt Weight of PRs
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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
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Phase Diagrams for The Chaos Synchronization
Symmetric coupling (?0)
Unidirectional coupling (?1)
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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
Blow-out Bifurcation
Riddling Bifurcation
Their Macroscopic Effects depend on The Existence
of The Absorbing Area.
? Supercritical case ? Appearance of An
Asynchronous Chaotic Attractor,
Exhibiting On-Off Intermittency ?
Subcritical case ? Abrupt Collapse
of A Synchronous Chaotic State
? Local riddling Attractor Bubbling ? Global
riddling Riddled Basin of Attraction
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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
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Symmetry-Conserving and Breaking Blow-out
Bifurcations
Linear Coupling
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
Dissipative Coupling
Hyperchaotic attractor for ?0
Chaotic attractor for ?1
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Classification of Periodic Orbits in Coupled 1D
Maps
For C0, 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, rq)
? Symmetric coupling (?0)
Conjugate orbit
rq/2 quasi-periodic transition to chaos Other
asynchronous orbits period-doubling transition
to chaos
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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
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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