Post on 20-Jan-2016
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Synchronization in Coupled Chaotic Systems
Sang-Yoon Kim
Department of Physics
Kangwon National University
Korea
Synchronization in Coupled Periodic Oscillators
Synchronous Pendulum Clocks Synchronously Flashing Fireflies
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Chaos and Synchronization
Lorenz Attractor [ Lorenz, J. Atmos. Sci. 20, 130 (1963)]
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)
z
yx
Butterfly Effect: Sensitive Dependence on Initial Conditions (small cause large effect)
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ChaoticSystem + Chaotic
System -
ts
ty ty ts
Secure Communication (Application)
Transmission Using Chaotic Masking
Transmitter Receiver
(Secret Message)
Several Types of Chaos Synchronization
Different degrees of correlation between the interacting subsystems
Identical Subsystems Complete Synchronization Nonidentical Subsystems Generalized Synchronization Phase Synchronization Lag Synchronization
Secret Message Spectrum
Chaotic MaskingSpectrum
Frequency (kHz)
4
21 1)( ttt Axxfx
An infinite sequence of period doubling bifurcations ends at a finite accumulation point 506092189155401.1A
When exceed , a chaotic attractor with positive Lyapunov exponent appears.
A
1210 tt xxxxxIterates: (trajectory) Attractor
tedtd )0()(
A
(x: seasonly breeding inset population)
1D Map (Building Blocks)
Complete Synchronization in Coupled Chaotic 1D Maps
Period-Doubling Transition to Chaos
5
).,()(
),,(1)(:
1
1
tttt
tttt
xygCyfy
yxgCxfxT
Coupling function
...,2,1)(),()(, nxxuxuyuyxg n
C: coupling parameter
Asymmetry parameter = 0: symmetric coupling exchange symmetry = 1: unidirectional coupling
Invariant synchronization line y = x
Synchronous orbits on the diagonal Asynchronous orbits off the diagonal
1.0,1 CA
22, xyyxg
Coupled 1D Maps
1
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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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WeakSynchronization
WeakSynchronization
StrongSynchronization
Transverse Bifurcations of UPOs
: Transverse Lyapunov exponent of the SCA (determining local transverse stability) 0 (SCA Transversely stable) Chaos Synchronization
(SCA Transversely unstable chaotic saddle) Complete Desynchronization
0
{UPOs} = {Transversely Stable Periodic Saddles (PSs)} + {Transversely Unstable Periodic Repellers (PRs)}
BlowoutBifurcation
FirstTransverseBifurcation
FirstTransverseBifurcation
BlowoutBifurcation
“Weight” of {PSs} > (<) “Weight” of {PRs} 00
0
0 0 0
0
C
Investigation of transverse stability of the SCA in terms of UPOs
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Strong Synchronization
All UPOs embedded in the SCA: Transversely stable
SCA: Asymptotically stable(Lyapunov stable + Attraction in the usual topological sense)
Attraction without bursting for all t
2,82.1 CA
e.g. Unidirectionally and Dissipatively Coupled Case with = 1 and g(x, y) = y2x2
Strong synchronization for A = 1.82 and Ct,l (= 2.789 …) < C < Ct,r (= 0.850 …)
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Global Effect of The First Transverse Bifurcation
Fate of A Locally Repelled Trajectory?
Attracted to another distant attractor
Transverse Bifurcation through which a first periodic saddle becomes transversely unstable
Local Bursting
Lyapunov unstable(Loss of Asymptotic
Stability)
Dependent on the existence of an Absorbing Area, controlling the global dynamics and acting as a bounded trapping area
Folding backof repelled trajectory
Local Stability Analysis: Complemented by a Study of Global Dynamics
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Bubbling Transition through The 1st Transverse Bifurcation
C
Strong synchronization BubblingRiddling
...789.2, ltC ...850.0, rtC
Case of rtCC ,Presence of an absorbing area Bubbling Transition
Noise and Parameter Mismatching Persistent intermittent bursting (Attractor Bubbling)
Transient intermittent bursting
Transcritical Contact Bif. Supercritical Period-Doubling Bif.
68.0,82.1 CA
005.0,68.0,82.1 CA
68.0,82.1 CA
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C
Strong synchronization BubblingRiddling
...789.2, ltC ...850.0, rtC
Case of ltCC ,
Transcritical Contact Bif. Supercritical Period-Doubling Bif.
Riddling Transition through The 1st Transverse Bifurcation
Disappearance of an absorbing area Riddling Transition
ltCC ,
an absorbing area surrounding the SCA
Contact between the SCA andthe basin boundary
ltCC ,67.2C
ltCC ,
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Riddled Basin
After the transcritical contact bifurcation, the basin becomes “riddled” with a dense set of “holes” leading to divergent orbits. The SCA is no longer a topological attractor; it becomes a Milnor attractor in a measure-theoretical sense.
As C decreases from Ct,l, the measure of the riddled basin decreases.
88.2C 93.2C
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Characterization of The Riddled Basin
Divergence Exponent
Divergence probability P(d) ~ d
(Take many randomly chosen initial conditions on the line y=x+d and determine which basin they lie in.) Measure of the Basin Riddling
Uncertainty Exponent
Uncertainty probability P() ~
(Take two initial conditions within a small square with sides of length 2 inside the basin and determine the final states of the trajectories starting with them.) Fine Scaled Riddling of the SCA
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Direct Transition to Bubbling or Riddling
Asymmetric systemsTranscritical bifurcation
Subcritical pitchfork or period-doubling bifurcation
Contact bifurcation(Riddling)
Non-contact bifurcation(Bubbling of hard type)
Symmetric systems(Supercritical bifurcations Bubbling transition of soft type)
Contact bifurcation(Riddling)
Non-contact bifurcation(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
Bubbling Riddling
Bubbling Riddling
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Basin Riddling through A Dynamic Stabilization
34.0,65.1 CA 26.0,65.1 CA
Symmetrically and dissipatively coupled case with =0 and 22, xyyxg
Bubbling Riddling
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Global Effect of Blow-out Bifurcations
C
Strong synchronization
rtC ,
BubblingRiddling
999.2, lbC 677.0, rbCltC ,
65.0,82.1 CA 65.0,82.1 CA
Weight of {PRs} > Weight of {PSs} SCA Transversely Unstable Chaotic Saddle Complete Desynchronization
Successive Transverse Bifurcations: Periodic Saddles (PSs) Periodic Repellers (PRs) (transversely stable) (transversely unstable)
For C < Cb,l, absence of an absorbing area Subcritical blow-out bifurcation Abrupt collapse of the synchronized chaotic state
For C > Cb,r, presence of an absorbing area Supercritical blow-out bifurcation Appearance of an asynchronous chaotic attractor covering the whole absorbing area and exhibiting the On-Off Intermittency
~ ~Blow-out Bif. First Transverse Bif. First Transverse Bif. Blow-out Bif.
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Symmetry-Conserving and -Breaking Blow-out Bifurcations Symmetrically and linearly coupled case with =0 and
Depending on the shape of a minimal invariant absorbing area, symmetry may or may not be conserved.
028.1,44.1
CA
xyyxg ),(
024.1,44.1
CA
031.1,427.1
CA
027.1,427.1
CA
Symmetry-Conserving Blow-out Bifurcation Symmetry-Breaking Blow-out Bifurcation
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Type of Asynchronous Attractors Born via Blow-out Bif.
{Asynchronous UPOs inside an absorbing area}={Asynchronous PSs with one unstable direction} +{Asynchronous PRs with two unstable directions}
Hyperchaotic attractor for =0 Chaotic attractor for =1
Weight of {PRs} > Weight of {PSs} Weight of {PRs} < Weight of {PSs}
333.0
,83.1
C
A
681.0
,83.1
C
A
Numbers of the period-11 saddles (Ns) and repellers (Nr):
Nr > Ns for < 0.8 Nr < Ns for > 0.9
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Phase Diagram for The Chaos Synchronization
Symmetric coupling (=0)
22, xyyxg Dissipatively coupled case with
Hatched Region: Strong Synchronization, Light Gray Region: Bubbling, Dark Gray Region: RiddlingSolid or Dashed Lines: First Transverse Bifurcation Lines, Solid Circles: Blow-out Bifurcation
Unidirectional coupling (=1)
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First TransverseBifurcation
Their Macroscopic Effects depend on The Existence of The Absorbing Area.
Blow-out Bifurcation
Investigation of The Mechanism for The Loss of Chaos Synchronization in terms of Transverse Bifurcations of UPOs embedded in The SCA (Periodic-Orbit Theory)
Attractor Bubbling
Basin Riddling
Subcritical case Abrupt Collapse of A Synchronous Chaotic State
Supercritical case Appearance of An Asynchronous Chaotic Attractor, Exhibiting The On-Off Intermittency.
ChaoticSaddle
Weakly-stableSCA
Strongly-stableSCA
Summary
References
[1] S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001). [2] S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 105, 187-196 (2001). [3] S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001).
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First TransverseBifurcation
Their Macroscopic Effects depend on The Existence of The Absorbing Area.
Blow-out Bifurcation
Investigation of The Mechanism for The Loss of Chaos Synchronization in terms of Transverse Bifurcations of UPOs embedded in The SCA (Periodic-Orbit Theory)
Attractor Bubbling
Basin Riddling
Subcritical case Abrupt Collapse of A Synchronous Chaotic State
Supercritical case Appearance of An Asynchronous Chaotic Attractor. The type (Symmetric or Asymmetric, Chaotic or Hyperchaotic) of which is determined by an absorbing area.
ChaoticSaddle
Weakly-stableSCA
Strongly-stableSCA
Summary
References
[1] S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001). [2] S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 105, 187-196 (2001). [3] S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001).
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Universality for The Chaos Synchronization
Mechanisms for The Loss of Chaos Synchronization in Coupled 1D Maps Are these mechanisms still valid for the real systems such as the coupled Hénon maps and coupled oscillators?
I think that those mechanisms are Universal ones, independently of the details of coupled systems, based on our preliminary results.
Universality for The Periodic Synchronization (well understood) The coupled 1D maps and coupled oscillators have the phase diagrams of the same structure and they exhibit the same scaling behavior on their critical set.
I believe that there may exist some kind of Universality for both the Chaotic and Periodic Synchronization in Coupled Dynamical Systems.
I suggest the Experimentalists to confirm this kind of universality in real experiment such as the electronic-circuit experiment.
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Phase Diagram for The Chaos Synchronization
Unidirectional coupling (=1)Symmetric coupling (=0)
xyyxg ,Linearly coupled case with
Hatched Region: Strong Synchronization, Light Gray Region: Bubbling, Dark Gray Region: RiddlingSolid or Dashed Lines: First Transverse Bifurcation Lines, Solid Circles: Blow-out BifurcationOpen Circles: Bdry. Crisis of An Absorbing Area, Open Squares: Bdry. Crisis of An Asyn. Chaotic Attractor
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Destruction of Hyperchaotic Attractorsthrough The Dynamic Stabilization
When a dynamic stabilization occurs before the blow-out bifurcation,a transition from bubbling to riddling takes place.
However, a sudden destruction of a hyperchaotic attractor occurswhen such a dynamic stabilization occurs after a blow-out bifurcation.
35.0,84.1 CA 328.0,84.1 CA
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Phase Diagram for Destruction of Hyperchaotic Attractors
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Phase Diagram for The Periodic Synchronization
Unidirectional coupling (=1)Symmetric coupling (=0)
22, xyyxg Dissipatively coupled case with
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Phase Diagram for The Periodic Synchronization
Unidirectional coupling (=1)Symmetric coupling (=0)
xyyxg ,Linearly coupled case with
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Effect of Parameter Mismatch and Noise for The Bubbling Case
Parameter mismatch or noise The SCA is broken up, and then it exhibits a persistent intermittent bursting. Attractor bubbling
01.0,75.0,82.1 CA
005.001.0
The maximum bursting amplitude increases when passing C=Ct,r.
bcbc
|y-x| max
|y-x| max
Ct,r Ct,r
005.0,75.0,82.1 CA
.1
,1:
2221
21
ttttt
tt
yxCyAy
AxxT
: Mismatching parameter
: Noise strength
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Abrupt Change of The Maximum Bursting AmplitudeThe maximum bursting amplitude increases abruptly through the interior crisis of the absorbing area for C 0.8437
Small absorbing area before the crisis Large absorbing area after the crisis
8438.0C 8.0C
Abrupt increase of the maximum bursting amplitude is in contrast to the case of symmetric coupling.
01.0Symmetric coupling (=0)
01.0
bc
|y-x| max
Unidirectional coupling (=1)|y-x| max
~
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Effect of Parameter Mismatch and Noise for The Riddling Case
0 ,91.2 ,82.1 CA 005.0
SCA with the riddled basin Chaotic transientParameter mismatch or noise
001.0
32
8.2C9228.1
: Average life-time of the chaotic transient
Exponential scaling(long lived chaotic transient)
Algebraic scalingC
~2/1
~ e
Ct,lC 2.84
Crossover
Characterization of The Chaotic Transients
8.2C9228.1
~2/1
~ e
~