Riddling Transition in Unidirectionally-Coupled Chaotic Systems
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Transcript of Riddling Transition in Unidirectionally-Coupled Chaotic Systems
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Riddling Transition in Unidirectionally-CoupledChaotic Systems
Sang-Yoon KimDepartment of PhysicsKangwon National UniversityKorea
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, Prog. Theor. Phys. 69, 32 (1983).]
z
yx
Butterfly Effect: Sensitive Dependence on Initial Conditions (small cause large effect)
• Other Pioneering Works • A.S. Pikovsky, Z. Phys. B 50, 149 (1984). • V.S. Afraimovich, N.N. Verichev, and M.I. Rabinovich, Radiophys. Quantum Electron. 29, 795 (1986). • L.M. Pecora and T.L. Carrol, Phys. Rev. Lett. 64, 821 (1990).
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Frequency (kHz)
Secret Message Spectrum
Chaotic MaskingSpectrum
ChaoticSystem + Chaotic
System -
ts
ty ty
ts
Secure Communication (Application)
Transmission Using Chaotic Masking
Transmitter Receiver
(Secret Message)
Several Types of Chaos SynchronizationDifferent degrees of correlation between the interacting subsystems Identical Subsystems Complete Synchronization [H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 (1983).]
Nonidentical Subsystems • Generalized Synchronization [N.F. Rulkov et.al., Phys. Rev. E 51, 980 (1995).]
• Phase Synchronization [M. Rosenblum, A.S. Pikovsky, and J. Kurths, Phys. Rev. Lett 76, 1804 (1996).]
• Lag Synchronization [M. Rosenblum, A.S. Pikovsky, and J. Kurths, Phys. Rev. Lett 78, 4193 (1997).]
[K.M. Cuomo and A.V. Oppenheim, Phys. Rev. Lett. 71, 65 (1993).]
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21 1)( ttt Axxfx
• Period-doubling transition to chaos An infinite sequence of period doubling bifurcations ends at a finite accumulation point A=1.401 155 189 092 506
1D Map (Building Blocks)
Chaos Synchronization in Unidirectionally Coupled 1D Maps
Unidirectionally Coupled 1D Maps
.),(
),,()(),(
:
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1
1
xyyxg
xygCyfyxfx
Ttttt
tt
• Invariant synchronization line y = x
Synchronous orbits on the diagonal Asynchronous orbits off the diagonal
1.0,1
CA
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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 (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)}
“Weight” of {PSs} > (<) “Weight” of {PRs} 00
Investigation of transverse stability of the SCA in terms of UPOs
Chaos Synchronization
BlowoutBifurcation
BlowoutBifurcation
0
0
0
C
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A Transition from Strong to Weak Synchronization
Weak SynchronizationWeak Synchronization Strong Synchronization
1st TransverseBifurcation
C
Attracted to another distant attractor
Dependent on the existence of an Absorbing Area, controlling the global dynamics and acting as a bounded trapping area
Folding backof repelled trajectory(Attractor Bubbling)
Local Stability Analysis: Complemented by a Study of Global Dynamics
(Basin Riddling)
1st TransverseBifurcation
• All UPOs embedded in the SCA: transversely stable PSs Strong Synchronization• A 1st PS becomes transversely unstable via a local Transverse Bifurcation.
Local Bursting Weak Synchronization
Fate of Local Bursting?
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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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Riddling Transition through A Transcritical Contact Bifurcation
Disappearance of An Absorbing Area through A Transcritical Contact Bifurcation
: saddle
: repeller
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C
Strong synchronization BubblingRiddling
...789.2, ltC ...850.0, rtC
Case of ltCC ,
Transcritical Contact Bif. Supercritical Period-Doubling Bif.
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 Probability P(d) Take many randomly chosen initial points on the line y=x+d and determine which basin they lie in Measure of the Basin Riddling
• Superpower-Law Scaling • Power-Law ScalingddP ~)(2/1
~)( dedP
CPower Law Superpower Law
Blow-out Bifurcation Riddling TransitionCrossover Region)81.284.2( C~ ~
874.1,795.2
C
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Uncertainty Exponent 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
• Superpower-Law Scaling • Power-Law Scaling ~)(P
2/1
~)( eP
0066.0,795.2
C
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Phase Diagram for The Chaotic and Periodic Synchronization
Hatched Region: Strong Synchronization, Light Gray Region: Bubbling, Dark Gray Region: RiddlingSolid or Dashed Lines: First Transverse Bifurcation Lines, Solid Circles: Blow-out Bifurcation
C0.0-0.8-2.6-3.4
1.4
1.6
1.8
2.0
A
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First TransverseBifurcation
Riddling transition occurs through a Transcritical Contact Bifurcation [S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001). S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 105, 187 (2001). ]
The same kind of riddling transition occurs also with nonzero (0 < 1) in general asymmetric systems [S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001).]
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)
ChaoticSaddle
Weakly-stableSCA
Strongly-stableSCA
Summary
Such riddling transition seems to be a “Universal” one occurring in Asymmetric Systems
).,()(),,(1)(
:1
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tttt
tttt
xygCyfyyxgCxfx
T
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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)
[Y.-C. Lai, C. Grebogi, J.A. Yorke, and S.C. Venkataramani, Phys. Rev. Lett. 77, 55 (1996).]
[S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001).]
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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[Y.L. Maistrenko, V.L. Maistrenko, O. Popovych, and E. Mosekilde, Phys. Rev. E 60, 2817 (1999).]
[V. Astakhov, A. Shabunin, T. Kapitaniak, and V. Anishchenko, Phys. Rev. Lett. 79, 1014 (1997).]
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( : constraint-breaking parameter)
Superpersistent Chaotic Transient
Parameter MismatchAB
02.0 and , ltCC
Average Lifetime: ( : some constants)21 & cc
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Chaotic Contact Bifurcation Saddle-Node Bifurcation (Boundary Crisis)
Transcritical Bifurcation Subcritical Pitchfork Bifurcation
2*)(
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ttxxC
t
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yyeByAxx
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ttxxC
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yyeByAxx
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x: Strongly unstable dir.y: Weakly unstable dir.
lineinvariant :00 y lineinvariant :00 y
Superpersistent Chaotic Transientaverage life time:
Superpersistent Chaotic Transient(Constraint-breaking: )
Superpersistent Chaotic Transient(Symmetry-breaking: )
21*
21
sBBcec 2
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cec 23
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cec
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ttxxC
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(x*: fixed point of the 1D map)
( : saddle-node bif. point)*sB