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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

1

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*)(

1

21

2

1:

ttxxC

t

tt

yyeByAxx

Tt

3*)(

1

21

2

1:

ttxxC

t

tt

yyeByAxx

Tt

x

y

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

12

1

cec 23

21

cec

2*)(

1

21

2

1:

ttxxC

t

tt

yyeByAxx

Tt

0 0

(x*: fixed point of the 1D map)

( : saddle-node bif. point)*sB