Chaos Synchronization in Coupled Dynamical Systems

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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
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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
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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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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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)

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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 gt (lt) Weight of PRs ?
Weak Synchronization
Weak Synchronization
Strong Synchronization
C
Blowout Bifurcation
First Transverse Bifurcation
First Transverse Bifurcation
Blowout Bifurcation
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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
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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
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Types of Asynchronous Attractors Born via
Blow-out Bifurcations
Chaotic attractor for unidirectional coupling
(?1)
Hyperchaotic attractor for symmetric coupling
(?0)
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Phase Diagrams for The Chaos Synchronization
Dissipatively coupled case with
Symmetric coupling (?0)
Unidirectional coupling (?1)
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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
Chaotic Saddle
Weakly-stable SCA
Strongly-stable SCA
Blow-out 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
? Attractor Bubbling ? Basin Riddling
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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)
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Transition from Bubbling to Riddling

• Boundary crisis of an absorbing area

• Appearance of a new periodic attractor inside
  the absorbing area

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