1

Synchronization in Coupled Chaotic Systems

Sang-Yoon Kim

Department of Physics

Kangwon National University

Synchronization in Coupled Periodic Oscillators

Synchronous Pendulum Clocks Synchronously Flashing Fireflies

2

Chaos and Synchronization Lorenz Attractor

[Lorenz, J. Atmos. Sci. 20, 130 (1963).]

Coupled Chaotic (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).

3

Frequency (kHz)

Secret Message Spectrum

Chaotic MaskingSpectrum

ChaoticSystem + Chaotic

System -

ts

ty ty

ts

Secure Communication (Application)

Encoding by Using Chaotic Masking

Transmitter Receiver

(Secret Message)

[K.M. Cuomo and A.V. Oppenheim, Phys. Rev. Lett. 71, 65 (1993).]

Decoding by Using Chaos Synchronization

ss

4

Several Types of Chaos Synchronization

Different 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).]

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21 1)( ttt axxfx

An infinite sequence of period doubling bifurcations ends at a finite accumulation point 506092189155401.1a

When a exceed a, a chaotic attractor with a positive Lyapunov exponent appears.

1210 tt xxxxxIterates: (trajectory) Attractor

tedtd )0()(

(x: seasonly breeding inset population)

1D Map (Representative model exhibiting universal scaling behavior)

Coupled 1D Maps

Period-Doubling Transition to Chaos

( > 0 chaotic attractor, < 0 regular attractor)

a aaa

6

).,()(

),,(1)(:

1

1

tttt

tttt

xygcyfy

yxgcxfxT

Coupling Function

...,2,1)(),()(, nxxuxuyuyxg n

c: coupling parameter

Asymmetry Parameter (0 1)

= 0: symmetric coupling exchange symmetry 0: asymmetric coupling ( = 1: unidirectional coupling)

Invariant Synchronization Line y = x

Synchronous Orbits on the Diagonal (, ) Asynchronous Orbits off the Diagonal ()

1.0,1 ca

22, xyyxg

Two Coupled 1D Maps

1

7

Transverse Stability of the Synchronous Chaotic Attractor

Synchronous Chaotic Attractor (SCA) on the Invariant Diagonal

SCA: Stable against the “Transverse Perturbation” Chaos Synchronization

An Infinite Number of Unstable Periodic Orbits (UPOs) Embedded in the SCA and Forming Its Skeleton Intimately Associated with the Transverse Stability of the SCA

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Chaos Synchronization

Competition between Periodic Saddles and Repellers

(: transverse Lyapunov exponent of the SCA)

{UPOs} = {Transversely Stable Periodic Saddles (PSs)} + {Transversely Unstable Periodic Repellers (PRs)}

BlowoutBifurcation

BlowoutBifurcation

Chaos Synchronization “Strength” of {PSs} > “Strength” of {PRs} < 0 (SCA: transversely stable)

0

0

0

c

Investigation of Transverse Stability of the SCA in Terms of UPOs

Complete Desynchronization “Strength” of {PSs} < “Strength” of {PRs} > 0 (SCA: transversely unstable chaotic saddle)

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Loss of Chaos Synchronization Unidirectionally Coupled 1D Maps (=1)

State Diagram

• An Infinite Number of UPOs inside the SCA.

).,(),(

),,(:

1

1

nnnn

nn

xycgayfy

axfxT

c: Coupling Parameter

222 ),( ,1),( xyyxgaxaxf

a=1.401 155 …

• Appearance of a Synchronous Chaotic Attractor (SCA) on the Invariant Diagonal when Passing a Critical Line (heavy solid line).

• Strong Synchronization (Hatched Region, <0) All Synchronous UPOs: Transversely Stable Periodic Saddles No Bursting (attracted to the diagonal without any bursting)• Transition to Weak Synchronization (Gray and Dark Gray Regions, <0) via a First Transverse Bifurcation of a Periodic Saddle Some UPOs: Transversely Unstable Local Bursting

a=1.82

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Fate of Local Bursting for the Case of Weak Synchronization

Fate of Local Bursting Dependent on the Global Dynamics

Attractor Bubbling (Gray Region)

Basin Riddling (Dark Gray Region)

Folded Back of a Locally Repelled Trajectory Transient Intermittent Bursting (<0)

Attracted to Another Distant Attractor

Basin of the SCA: Riddled with a Dense Set of “Holes,” Leading to Another Attractor

WeakSynchronization

(Riddling)

WeakSynchronization

(Bubbling)

StrongSynchronization

BlowoutBifurcation

BlowoutBifurcation

FirstTransverseBifurcation

FirstTransverseBifurcation

• Weak Synchronization: Some UPOs: Transversely Unstable Local Bursting

c

11

Transcritical Transition to Basin Riddling

Riddling Transition (Basin: riddled with a dense set of “holes”)

Contact between the SCA and the basin boundary

an absorbing area surrounding the SCA

(: repeller on the basin boundary : saddle on the diagonal)

67.2c ...789.2c

93.2c

[S.-Y. Kim and W. Lim, Phys. Rev. E 63, 026217 (2001). S.-Y. Kim and W. Lim, Prog. Theor. Phys. 105, 187 (2001). S.-Y. Kim and W. Lim, Phys. Rev. E 64, 016211 (2001).]

Riddled Basin

88.2c After the transcritical bifurcation, the basin becomes “riddled” with a dense set of “holes” leading to divergent orbits.

As c decreases from ct,l, the basin riddling is intensified.

c

Strong synchronization BubblingRiddling

...789.2, ltc ...850.0, rtc

Transcritical Bif. Supercritical Period-Doubling Bif.

Transcritical Bifurcation (Stability Exchange)

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Characterization of the Riddled Basin Divergence Exponent Divergence Probability P(d) ~ d

Uncertainty Exponent Uncertainty Probability P() ~

Measure of the Basin Riddling

Two Initial Condition: Uncertain if their final states are different Fine Scaled Riddling of the SCA

[c Blowout Bifurcation Point cb,l (=-2.963) () P(d): Increase]

[c cb,l () P(): Increase]

c

c

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Effect of Parameter Mismatching on Weak Synchronization[A. Jalnin and S.-Y. Kim, Phys. Rev. E 65, 026210 (2002). S.-Y. Kim, W. Lim, and Y. Kim, Prog. Theor. Phys. 107, 239 (2002).]

Unidirectionally Coupled Nonidentical 1D Maps

).(),(

),,(: 22

1

1

nnnn

nn

xycayfy

axfxT

: Mismatching Parameter

All UPOs (embedded in the SCA): Transversely Stable No Parameter Sensitivity

Riddling

Effect of Parameter Mismatching (=0.001)

91.2c

1.2c

Bubbling

7.0c

(1) Strong Synchronization (Small Mismatching Small Effect)

(2) Weak Synchronization (Small Mismatching Large Effect)

SCA with a Riddled Basin (Gray) Chaotic Transient (Black) with a Finite Lifetime

SCA Bubbling Attractor(exhibiting persistent intermittent bursting)

Slightly Perturbed SCA ( Mismatching Strength)

Local Transverse Repulsion of Periodic Repellers Parameter Sensitivity

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Characterization of the Parameter Sensitivity of a Weakly Stable SCA

Characterization of Parameter Sensitivity• Measured by Calculating a Derivative of the Transverse Variable un (=xn-yn) with respect to the Mismatching Parameter along a Synchronous Trajectory

Representative Value (by Taking the Minimum Value of in an Ensemble of Randomly Chosen Initial Orbit Points)

)( *0xN

)(min *0*

0

xNx

N Parameter Sensitivity Function:

• Strong Synchronization (SS) N: Bounded No Parameter Sensitivity

• Weak Synchronization (WS) N ~ N: Unbounded Parameter Sensitivity : Parameter Sensitivity Exponent (PSE) Used to Measure the Degree of Parameter Sensitivity

c=-0.7 (WS)

c=-1.5 (SS)

.),()(1

*1

*0

0

N

kkaN

N axfxSu

)( *

kkN xR

1

0

** )](),([M

iikikx xchaxf)( *

kM xR )exp( MM

Exponent Lyapunov Transverse time)-(M Local :M

MultiplierStability Transverse time)-(M Local :)( *kM xR

0]for Orbit sSynchronou :)},[{ *** kkk xyx

7.0cIntermittent Behavior

Boundedness of SN

|)(|max)( *0

0

*0 xSx k

NkN

Looking only at the Maximum Values of |SN|:

]),(,2)(,2),([ 2xaxfxxhaxaxf ax

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Characterization of the Bubbling Attractor and the Chaotic Transient

Parameter Sensitivity Exponents (PSEs)

• Monotonic Increase of () as c is Changed toward the Blowout Bifurcation Point

(Due to the Increase in the Strength of Local Transverse Repulsion of the Embedded Periodic Repellers.)

Scaling for the Average Characteristic Time

~~ *uNN

() =1/ ()

~ 1/

Average Laminar Length (Interburst Interval) of the Bubbling Attractor: ~ -

Average Lifetime of Chaotic Transient: ~ -

Reciprocal Relation between the Scaling Exponent and the PSE

State Bursting|| State,Laminar value)threshold(|| ** bnbn uuuu

Escaped have as Regarded value)threshold(|| * cn uu

a=1.82, c=-0.7, =0.001

0 1500 3000

0.8

0.0

-0.8

n

u n

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Effect of Noise on Weak Synchronization

Characterization of the Noise Sensitivity (=0.0005)

;)()(1

1**

00

N

kkkkNN

N xRxSu

NxSnNnx

N ~|)(|maxmin *0

0*0

Unidirectionally Coupled Noisy 1D Maps

.)()(

,)(:

)2(221

)1(1

nnnnn

nnn

xycyfy

xfxT

[S.-Y. Kim, W. Lim, A. Jalnin, and S.-P. Kuznetsov, Phys. Rev. E 67, 016217 (2003).]

7.0c

Exponent Lyapunov Transverse time)-(M Local:M

: Bounded Noise → Boundedness of SN: Determined by RM (same as in the parameter mismatching case)

[Noise Sensitivity Exponent() = PSE()]

Noise Effect = Parameter Mismatching Effect

Characterization of the Bubbling Attractor and the Chaotic Transient

~ - ; () =1/ ()

91.2c

Bubbling Attractor Chaotic Transient

(: average time spending near the diagonal)

)exp()(, *)2()1( MmMnnn MxR

Strength Noise :

nceunit varia a andmean zero a with variablerandom Uniform:)2,1()( ii

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Dynamical Consequence of Blowout Bifurcations

: parameter tuning the degree of asymmetry of the coupling =0 Symmetric Coupling Case, =1 Unidirectional Coupling Case

.10)],()([)(

)],()([)1()(:

1

1

nnnn

nnnn

yfxfcyfy

xfyfcxfxT

[S.-Y. Kim, W. Lim, E. Ott, and B. Hunt, Phys. Rev. E 68, 066203 (2003).]

Two Coupled 1D maps

Asynchronous Hyperchaotic Attractor with 2>0 for =0

Asynchronous Chaotic Attractor with 2<0 for =1

97.1a

2283.0s

parameter coupling scaled :])2/1([ cs

2283.0s

Appearance of an Asynchronous Attractor via a Blowout Bifurcation of the SCA

cBlowout

BifurcationBlowout

Bifurcation

Synchronization

0 00

0028.0

6157.0

2

1

0024.0

6087.0

2

1

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Mechanism for the Appearance of the Asynchronous Hyperchaotic and Chaotic Attractors

Decomposition of the 2nd Lyapunov Exponent 2

d = |v| [|(x-y)/2|]: Transverse Variable d < d*: Laminar Component (Off State) d > d*: Bursting Component (On State)

)0( || 222222 llbbl

0with Attractor Chaotic usAsynchrono

0with Attractor icHyperchaot usAsynchrono

0such that 8520 Value Threshold

222

222

222*

bl

bl

bl

||

||

:~~||).~(

*

*

Competition between the Laminar and Bursting Components

d (t)

: =0, : =0.852 : =1

d* :)(2

bl Weighted 2nd Lyapunov Exponent for the Laminar (Bursting) Component

Sign of 2 (= b2 |l

2|): Determined by the Competition between the Laminar and Bursting Components

Intermittent Asynchronous Attractor Born via a Blowout Bifurcation

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Partial Synchronization in Three Coupled Chaotic Systems

Fully Synchronized Attractor (FSA) for the Case of Strong Coupling

Breakdown of the Full Synchronization via a Blowout Bifurcation

Partial Synchronization (PS) Complete Desynchronization

)( )3()2()1( xxx

),( )3()2()2()1( xxxx )( )3()2()1( xxx : Two-Cluster State for p=0

)3,2,1()],()([)( )(3

1

)()()(1

ixfxfpcxfx i

nj

jnj

in

in

Three Coupled 1D Maps

coupling. Symmetric :3/1 coupling, ional Unidirect:0),3/10( :Path

).1 (element th for the weight coupling :)/(

32

3

1

pppppp

pjNNpj jjj

Occurrence of the Partial Synchronization

3/1p

[W. Lim and S.-Y. Kim, Phys. Rev. E 71, 035221 (2005).]

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Transverse Stability of Two-Cluster States

Unidirectional Coupling Case Symmetric Coupling Case

nnnnn YxxXx )3()2()1( ,

)].()([)21()()],()([2)( 11 nnnnnnnn YfXfcpYfYXfYfpcXfX

Reduced 2D Map Governing the Dynamics of a Two-Cluster StateTwo-Cluster State:

Threshold Value p* (~ 0.146) s.t. • 0p<p* Two-Cluster State: Transversely Stable (<0) Occurrence of the PS

• p*<p1/3 Two-Cluster State: Transversely Unstable (>0) Occurrence of the Complete Desynchronization

Intermittent Two-Cluster State Born via the Blowout Bifurcation of the FSA

0p 3/1p

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Mechanism for the Occurrence of the Partial Synchronization Decomposition of the Transverse Lyapunov Exponent

d = |V| [V(X-Y)/2]: Transverse Variable d < d*: Laminar Component, d > d*: Bursting Component.

)0( || llbbl

d (t)

)

||/

)||

:~~||).~(

*

*

0(ization Desynchron Complete theof Occurrence

ly UnsableTransverse :StateCluster -Two31

0( PS theof Occurrence Stablely Transverse :StateCluster -Two0

0such that 1460 Value Threshold *

bl

bl

bl

pp

pp

p

Competition between the Laminar and Bursting Components

: p=0, : p=0.146: p=1/3

d* :)(bl Weighted Transverse Lyapunov Exponent for the

Laminar (Bursting) Component

Sign of (= b |l

|): Determined by the Competition between the Laminar and Bursting Components

Intermittent Two-ClusterState Born via a Blowout Bifurcation

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Summary

1. Transcritical Transition to Basin Riddling in Asymmetrically Coupled Chaotic Systems

2. Characterization of the Parameter Mismatching and Noise Effects on Weak Synchronization in terms of the PSEs and NSEs

3. Investigation of Dynamical Origin for the Occurrence of Hyperchaos and Chaos via Blowout Bifurcation through Competition between the Laminar and Bursting Components

Weak Synchronization(Riddling)

Weak Synchronization(Bubbling)Strong Synchronization

BlowoutBifurcation

BlowoutBifurcation

First TransverseBifurcation

First TransverseBifurcation

00

0,0 0,0 0,0 c

Investigation of Loss of Chaos Synchronization in Two Coupled Chaotic Systems

1. Investigation of the Dynamical Mechanism for the Occurrence of the Partial Synchronization through Competition between the Laminar and Bursting Components

Investigation of Loss of Full Synchronization in Three Coupled Chaotic Systems

Full SynchronizationPartial Synchronization (Clustering)

Complete Desynchronization