1

Mechanism for the Partial Synchronization in Coupled Logistic Maps

Cooperative Behaviors in Many Coupled Maps

Woochang Lim and Sang-Yoon KimDepartment of PhysicsKangwon National University

Fully Synchronized Attractor for the Case of Strong Coupling

Breakdown of the Full Synchronization via a Blowout Bifurcation

Partial Synchronization (PS) Complete Desynchronization

)( 321 XXX

),( 3221 XXXX )( 321 XXX : Clustering

2

N Globally Coupled 1D Maps

)...,,1(1)(,))(())(()1()1( 2

1

NiaxxftxfN

ctxfctx

N

jjii

Reduced Map Governing the Dynamics of a Three-Cluster State

)3,2,1(,))(())(()1()1(3

1

itXfpctXfctXj

jjii

.)()(,)()(,)()( 321 121211111XtxtxXtxtxXtxtx

NNNNNNN iiiiii

pi (=Ni/N): “coupling weight factor” corresponding to the fraction of

the total population in the ith cluster

3

11

j jp

Three-Cluster State

Coupled Logistic Maps (Representative Model)

Reduced 3D Map Globally Coupled Maps with Different Coupling Weight

Investigation of the PS along a path connecting the symmetric and unidirectional coupling cases: p2=p3=p, p1=1-2p (0 p 1/3)

p1=p2=p3=1/3 Symmetric Coupling Case No Occurrence of the PSp1=1 and p2=p3=0 Unidirectional Coupling Case Occurrence of the PS

3

Transverse Stability of the Fully Synchronized Attractor (FSA)

• Longitudinal Lyapunov Exponent of the FSA

M

tt

MXf

M 1

*|| |)('|ln

1lim

• Transverse Lyapunov Exponent of the FSA

For c>c* (=0.4398), <0 FSA on the Main Diagonal

Occurrence of the Blowout Bifurcation for c=c*

• FSA: Transversely Unstable (>0) for c<c*

• Appearance of a New Asynchronous Attractor

Transverse Lyapunov exponent

a=1.95

2)tymultiplici(|1|ln || c

a=1.95, c=0.5 a=1.95, c=0.5

*

321 )()()(

tX

tXtXtX

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Type of Asynchronous Attractors Born via a Blowout Bifurcation

Unidirectional Coupling Case (p=0)Two-Cluster State: Transversely Stable Partially Synchronized Attractor on the 23 Plane Occurrence of the PS

010.0~

,021.0~

,539.0~

42.0,95.1

3

2

1

ca

Symmetric Coupling Case (p=1/3)

014.0~

,014.0~

,579.0~

42.0,95.1

3

2

1

ca

Appearance of an Intermittent Two-Cluster State on the Invariant 23 Plane ({(X1, X2, X3) | X2=X3}) through a Blowout Bifurcation of the FSA

Two-Cluster State: Transversely Unstable Completely Desynchronized (Hyperchaotic) Attractor Filling a 3D Subspace (containing the main diagonal) Occurrence of the Complete Desynchronization

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Two-Cluster States on the 23 Plane

*32

*1 )()(,)( tt YtXtXXtX

)].()([)21()()],()([2)( ****1

****1 tttttttt YfXfcpYfYXfYfpcXfX

2,

2

**** YXV

YXU

.)1(2,)41(2)(1 122

1 tttttttt VUcaVVUpacVUaU

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

For numerical accuracy, we introduce new coordinates:

Two-Cluster State:

003.0c003.0c

Unidirectional Coupling Case Symmetric Coupling Case

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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

~

Transverse Stability of Two-Cluster States

0p

3/1p

146.0p

95.1a

M

ttt

MVUf

Mc

1

|)('|ln1

lim|1|ln

Transverse Lyapunov Exponent of the Two-Cluster State

(c cc*)

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Mechanism for the Occurrence of the Partial Synchronization

Intermittent Two-Cluster State Born via a Blowout Bifurcation

Decomposition of the Transverse Lyapunov Exponent of the Two-Cluster State

)()(

)( blbl

bl

:),( bliL

Li

i Fraction of the Time Spent in the i Component (Li: Time Spent in the i Component)

Transverse Lyapunov Exponent of the i Component(primed summation is performed in each i component)

: Weighted Transverse Lyapunov Exponent for the Laminar (Bursting) Component

:|)(')1(|ln'1

state

ittti

i VUfcL

)0( || llbbl

d = |V|: Transverse Bursting Variabled*: Threshold Value s.t. d < d*: Laminar Component (Off State), d > d*: Bursting Component (On State).

We numerically follow a trajectory segment with large length L (=108), and calculate its transverse Lyapunov exponent:

1

0

|)(')1(|ln1 L

ttt VUfc

L

d (t)

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Threshold Value p* ( 0.146) s.t. :0~~|| bl

bl ||0p<p *

p*<p1/3

Two-Cluster State: Transversely Stable Occurrence of the PS

Sign of : Determined via the Competition of the Laminar and Bursting Components

bl ||

Two-Cluster State: Transversely Unstable Occurrence of the Complete Desynchronization

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

Competition between the Laminar and Bursting Components

Laminar Component

Bursting Componentpp l

lll

l oftly independen same, Nearly the :)( oft independenNearly :and

ppp bb

bbb increasingh Larger wit :)( increasingh Larger wit :, oft independenNearly :

~

|)|( lb

a=1.95, d*=10-4

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Mechanism for the Occurrence of the Partial Synchronization in Coupled 1D Maps

Sign of the Transverse Lyapunov Exponent of the Two-Cluster State Born via a Blowout Bifurcation of the FSA: Determined via the Competition of the Laminar and Bursting Components

Summary

)0(|| bl Two-Cluster State: Transversely Stable Occurrence of the PS

)0(|| bl Two-Cluster State: Transversely Unstable Occurrence of the Complete Desynchronization

|]|[ lb