11

Dynamical Routes to Clusters and Scaling in Globally Coupled Chaotic Systems

Sang-Yoon Kim

Department of Physics

Kangwon National University

Globally Coupled Systems (Each element is coupled to all the other ones with equal strength)

Biological Examples Heartbeats, Circadian Rhythms, Brain Rhythms, Flashing of Fireflies

Nonbiological Examples Josephson Junction Array, Multimode Laser, Electrochemical Oscillator

Incoherent State

(i: index for the element, : Ensemble Average)

(each element’s motion: independent) (collective motion)Coherent State

Stationary Snapshots Nonstationary Snapshots

t = n t = n+1

Synchronized Flashing of Fireflies

222

Emergent Science

“The Whole is Greater than the Sum of the Parts.”

Complex Nonlinear Systems: Spontaneous Emergence of Dynamical Order

Order Parameter ~ 0 Order Parameter < 1 Order Parameter ~ 1

3

Two Mechanisms for Synchronous Rhythms

Leading by a Pacemaker Collective Behavior of All Participants

44

Synchronization of Pendulum Clocks

Synchronization by Weak Coupling Transmitted through the Air or by Vibrations in the Wall to which Theyare Attached

First Observation of Synchronization by Huygens in Feb., 1665

55

Circadian Rhythms

Biological Clock Ensemble of Neurons in the Suprachiasmatic Nuclei (SCN) Located within the Hypothalamus: Synchronization → Circadian Pacemaker

[Zeitgebers (“time givers”): light/dark]

Time of day (h) Time of day (h)

Tem

pera

ture

(ºc

)

Gro

wth

Hor

mon

e (n

g/m

L)

66

Integrate and Fire (Relaxation) Oscillator Mechanical Model for the IF Oscillator

Van der Pol (Relaxation) Oscillator

Accumulation (Integration) “Firing”

wat

er l

evel

wat

er o

utfl

ow

time

time

Firings of a Neuron Firings of a Pacemaker Cell in the Heart

77

Synchronization in Pulse-Coupled IF Oscillators

Population of Globally Pulse-Coupled IF Oscillators

)(any for )N

min(1,:Coupling Pulse

ResettingFiring: of Value Threshold

ij(t)x)t(x)t(x

vv

.N,...,i,v,Ivdtdv

jii

ii

iii

1

01

110

Full Synchronization

Kicking

Heart Beat: Stimulated by the Sinoatrial (SN) Node Located on the Right Atrium, Consisting of Pacemaker Cells 410

20

2

.

I

[R. Mirollo and S. Strogatz, SIAM J. Appl. Math. 50, 1645 (1990)]

[Lapicque, J. Physiol. Pathol. Gen. 9, 620 (1971)]

8

Emergence of Dynamical Order and Scaling in A Large Population of Globally Coupled Chaotic Systems

Scaling Associated with Clustering (Partial Synchronization)

Successive Appearance of Similar Clusters of Higher Order

99

Period-Doubling Route to Chaos

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

Butterfly Effect [Small Cause Large Effect] Sensitive Dependence on Initial Conditions

Logistic Map[May, Nature 261, 459 (1976)]

21 1)( ttt axxfx : Representative Model for Period-Doubling Systems

a*a

: Lyapunov Exponent (exponential divergence rate of nearby orbits) 0 Regular Attractor > 0 Chaotic Attractor

a*a

Transition to Chaos at a Critical Point a* (=1.401 155 189 …) via an Infinite Sequence of Period Doublings

1010

Universal Scaling Associated with Period Doublings Logistic Map

2

11)(

tttaxxfx

Parametrically Forced Pendulum

[ M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978),]

Universal Scaling Factors: =4.669 201 … =-2.502 987 …

.2sin)2cos(22),,(),,,(, 2

11111xtAytyxftyxfyyx

[S.-Y. Kim and K. Lee, Phys. Rev. E 53, 1579 (1996).]

h(t)=Acos(2t)

(=0.7, =1.0, A* =2.759 832)

1111

Globally Coupled Chaotic Maps

An Ensemble of Globally Coupled Logistic Maps

....,,1];))(())((1

[))(()1(1

NitxftxfN

txftxi

N

jjii

• A Population of 1D Chaotic Maps Interacting via the Mean Field:

N

jjj txf

Ntxfth

1

))((1

))(()(

21)( axxf

• Dissipative Coupling Tending to Equalize the States of Elements

Investigation: Scaling Associated with Emergence of Clusters

Main InterestOccurrence of Clustering (Appearance of Clusters with Different Synchronized Dynamics)

[Experimental Observations: Electrochemical Oscillators, Salt-Water Oscillators, Belousov-Zhabotinsky Reaction, Catalytic CO Oxidation]

1212

tN Xtxtx )()(1

Fully Synchronized Chaotic Attractor on the Invariant Diagonal

Complete Chaos Synchronization

Transverse Lyapunov Exponent of the FSA

1D Reduced Map Governing the Dynamicsof the Fully Synchronized Attractor (FSA):

M

tt

MXf

M 1

|)('|ln1

lim|1|ln

21 1)( ttt aXXfX

: Transverse Lyapunov exponent associated with perturbation transverse to the diagonal

For strong coupling, < 0 Complete Synchronization

For < *(~ 0.2901), > 0 FSA: Transversely Unstable Transition to Clustering State

a=0.15

a=0.15

1313

)].([ cluster 2nd :)()(

),( cluster 1st :)()()(

121

121

1

1

NNNYtxtx

NXtxtxtx

tNN

tN

Two-Cluster States on an Invariant 2D Plane

Two-Cluster States

Transverse Lyapunov Exponents

,1 (,2): Transverse Lyapunov exponent associated with perturbation

breaking the synchrony of the 1st (2nd) cluster,1<0 and ,2<0 Two-cluster state: Transversely Stable Attractor in the original N-D state space

2D Reduced Map Governing the Dynamicsof the Two-Cluster State:

)].()([)1()(

)],()([)(

1

1

tttt

tttt

YfXfpYfY

XfYfpXfX

M

tt

M

M

tt

MYf

MXf

M 12,

11, |)('|ln

1lim|1|ln,|)('|ln

1lim|1|ln

p (=N2/N): Asymmetry Parameter (fraction of the total population of elements in the 2nd cluster)

0 (Unidirectional coupling) < p 1/2 (Symmetric coupling)

a=0.15=0.05

1414

Classification of Periodic Orbits in Terms of the Period and Phase Shift (q,s)

Scaling Associated with Periodic Orbits for the Two-Cluster Case

Scaling near the Zero-Coupling Critical Point (a*, 0) for p=1/2

q different orbits with period q distinguished by the phase shift s (=1,…,q-1) in the two uncoupled (=0) logistic maps

(Synchronous) In-phase orbit on the diagonal (s = 0) (Asynchronous) Anti-phase (180o out-of-phase) orbit with time shift of half a period (s = q/2)(Asynchronous) Non-antiphase orbits (Other s) Two orbits with phase shifts s and q- s: Conjugate-phase orbits (under the exchange X Y for p=1/2)

0,,4 * aaq

Stability Diagrams of the Conjugate-Phase Periodic Orbits

[1. S.P.Kuznetsov, Radiophysics and Quantum Electronics 28, 681 (1985). 2. S.-Y. Kim and H. Kook, Phys. Rev. E 48, 785 (1993). 3. S.-Y. Kim and H. Kook, Phys. Lett. A 178, 258 (1993).]

Renormalization Results: Scaling Factor for the Coupling Parameter =2 [i.e., ’(=2)]

1515

Dynamical Routes to Two-Cluster States (p=1/2)

Dynamical Route to Two-Cluster State for a=0.15 (Two Stages)

FSA (Strong coupling) Blowout Bifurcation (*=0.2901) (“Complex” Gray line)Transversely Unstable

(1) Jump to Anti-phase Period-2 Two-Cluster State

Complete Synchronization

Stabilization of anti-phase period-2 attractor via subcritical pitchfork bifurcation

(2) Transition to Conjugate-Phase Period-4 Two-Cluster States

For < 0.0862, Two-Cluster Chaotic State: Transversely Unstable (Gray dots) High-Dimensional State

3.0,15.0 a

11.0,15.0 a105.0

15.0

aYXV

1616

Scaling for the Dynamical Routes to Clusters (p=1/2)

Successive Appearance of Similar Cluster States of Higher Orders

(1) 1st-Order Renormalized State

(2) 2nd-Order Renormalized State

As the zero-coupling critical point (a*, 0) is approached, similar cluster states of higher orders appear successively.

2/3.0,/15.0 a

1717

Effect of Asymmetric Distribution of Elements

p (asymmetry parameter): smaller

Conjugate-Phase Two-ClusterStates: Dominant

Appearance of Similar Cluster States

2/15.0 a/15.0a

15.0a 3.0,15.0 pa

1818

System

Clustering in the Linearly Coupled Maps

N

jjtitii tx

NMtxMtxftx

1

)(1

)];([))(()1(

Governing Eqs. for the Two-Cluster State

).()1()(),()( 11 tttttttt YXpYfYXYpXfX

).Y)t(x)t(x),X)t(x)t(x)t(x tNNN cluster (2nd cluster(1st 1121 11

Scaling for the Linear Coupling Case (P=1/2)

[ for the inertial coupling case; 2 for the dissipative coupling case]

[ Renormalization Results: 1. S.P.Kuznetsov, Radiophysics and Quantum Electronics 28, 681 (1985). 2. S.-Y. Kim and H. Kook, Phys. Rev. E 48, 785 (1993). 3. S.-Y. Kim and H. Kook, Phys. Lett. A 178, 258 (1993).]

[Linear Mean Field ‘Inertial Coupling’ (each element: maintaining the memory of its previous states)]

:))((1

cf.[1

N

jjt txf

NM Nonlinear Mean Field Dissipative Coupling

(Tendency of equalizing the states of the elements)]

1919

Successive Appearance of Similar Cluster States of Higher Ordersfor the Linear Coupling Case (P=1/2)

(1) 0th-Order Cluster State

(2) 1st-Order Renormalized State

(3) 2nd-Order Renormalized State

(No Complete Chaos Synchronization near the Zero Coupling Critical Point)

2/15.0 a

/6.0,/15.0 a

6.0,15.0 a 15.0a

/15.0a

2020

Asymmetric Effect on the Dynamical Routes to Clusters

p (asymmetry parameter): smaller

Conjugate-Phase Two-ClusterStates: Dominant

Appearance of Similar Cluster States2/15.0 a/15.0a

15.0a 35.0,15.0 pa

(Similar to the Dissipative Coupling Case)

2121

Dynamical Routes to Clusters and Scaling in Globally Coupled Oscillators

(Purpose: to examine the universality for the results obtained in globally coupled maps)

Globally Coupled Parametrically Forced Pendula (Dissipative Coupling)

.2sin)2cos(22),,(

],1

[),,(,

2

1

xtAytyxf

yyN

tyxfyyx i

N

jjiiiii

Governing Eqs. for the Dynamics of the Two-Cluster States

Cluster 2nd

Cluster1st

:)t(Y)t(y...)t(y),t(X)t(x...)t(x

:)t(Y)t(y...)t(y),t(X)t(x...)t(x

NNNN

NN

2121

1111

11

11

).()1(),,(,),(),,(, 21222221211111 YYptYXfYYXYYptYXfYYX

Scaling for the Conjugate-Phase Periodic Orbits

2.66 2.74 2.820.00

0.05

0.10

A

x

1

7.0

Period-Doubling Route to Chaos in the Single Pendulum (A*=2.759 833)

22

Similar Cluster States for the Dissipative Coupling Case

(1) 0th-Order Cluster State

(2) 1st-Order Renormalized State

(3) 2nd-Order Renormalized State 2/016.0 A

/016.0A

)(016.0 211 XXVA 016.0A

/016.0A

2/016.0 A

2323

Similar Clusters for the Inertial Coupling Case

(2) 1st-Order Renormalized State /016.0A

System ].1

[),,(,1

N

jijiiiii xx

Ntyxfyyx

Appearance of Similar Cluster States (Scaling for the Coupling Parameter: )

/016.0A

(1) 0th-Order Cluster State

(3) 2nd-Order Renormalized State

016.0A 016.0A

2/016.0 A 2/016.0 A

2424

Similar Clusters in Globally Coupled Rössler Oscillators

(1) 0th-Order Cluster State

(2) 1st-Order Renormalized State/33.0c

33.0c 33.0c

/33.0c

Globally Coupled Rössler Oscillators (Dissipative Coupling)

Appearance of Similar Cluster States

).(,],1

[1

cxzbzayxyxxN

zyx iiiiiii

N

jjiii

232204.4,2.0 * cba

2525

Summary

Investigation of Dynamical Routes to Clusters in Globally Coupled Logistic Maps

Universality for the Results

Confirmed in Globally Coupled Pendulums

(a, c) (a*, 0): zero-coupling critical point

Successive Appearance of Similar Cluster States of Higher Orders

Our Results: Valid in Globally Coupled Period-Doubling Systems of Different Nature