1

Emergence of Coherence and Scaling in an Ensemble of 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

22

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

4

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)

6

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

Van der Pol (Relaxation) Oscillator

Accumulation (Integration) “Firing”

wat

er le

vel

wat

er o

utfl

ow

time

time

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

7

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 Coherence with a Macroscopic

Mean Field Successive Appearance of Similar

Coherent States 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

10

Universal Scaling Associated with Period Doublings Noisy Logistic Map

;)(1 ttt xfx

Parametrically Forced Pendulum

[ M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978), J. Crutchfield, M. Nauenberg, and J. Rudnick, Phys. Rev. Lett. 46, 459 (1981). B. Shraiman, C.E. Wayne, and P.C. Martine, Phys. Rev. Lett. 46, 462 (1981).]

nce.unit varia a andmean zero a with noise random uniform :)(

strength, noise:

t

310 /10 3 Universal Scaling Factors: =4.669 201 … =-2.502 987 … =6.619 03

w: Gaussian White Noise with <w(t)>=0 and <w(t1) w(t2)> = (t1 – t2).

.2sin)2cos(22),,(),),,(, 211111 xtAytyxfttyxfyyx w

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

-A*

h(t)=Acos(2t)

(=2, =1, A* = 6.57615 …)

11

Globally Coupled Noisy Chaotic Systems

An Ensemble of Globally Coupled Noisy Logistic Maps

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

[)())(()1(1

NitxftxfN

ttxftx i

N

jjiii

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

N

jjj txf

Ntxfth

1

))((1

))(()(

:

:)t(;ax)x(f

21

• Dissipative Coupling Tending to Equalize the States of Elements

Investigation: Coherence in Asynchronous Chaotic Attractors and Scaling Associated with the Mean Field

Uniform Random Noise with a Zero Mean and Unit Variance

Parameter to Control the Noise Strength

Main Interest

12

Scaling near the Zero-Coupling Critical Point (N=10 ) Universal Scaling near the Zero-Coupling Critical Point (a, , ) = (a*, 0, 0): (a* = 1.401 155 189 …)

Dynamical Behavior at a Set of Parameters (a, , ) Dynamical Behavior [with Doubled Time Scale (t=2)] at a Set of Renormalized Parameters (a/, / c, /) [=4.669201, =6.61903]

Asynchronous Chaotic Attractors (containing the diagonal)

“Scaling Factor for the Coupling Parameter for the Dissipative Coupling:

/.

,/.,/.a

0010

2020320

0010

020320

.

,.,.a

2

22

0010

2020320

/.

,/.,/.a

[ 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).]

2c

4

13

Ensemble-Averaged Mean Field h(t)

)(ofAverageTime:)(1

lim)(,1

)(1

T1

ththT

th(t))f(xN

thT

t

n

jj

ary)Nonstation(

)Stationary:(

:h(t))t(h)t(h

)t(h)t(h)t(h

In the Thermodynamic Limit of N→∞,

Coherent and Incoherent States

Incoherent State (Each Element: Independent Motion)

Coherent State (Collective Motion)

Multiple Transitions to Coherence for 0010320104 ..aN and,,

Incoherent State (Gray, ε=0.02)Coherent State (Black, ε=0.03)

14

Mean- Field Dynamic of Asynchronous Chaotic Attractors for Δa=0.32

ε=0.02 ε=0.03 ε=0.03

ε=0.033 ε=0.035 ε=0.036

ε=0.036ε=0.035ε=0.033

15

Order Parameter for the Coherent Transition

Order Parameter

:))()(())(( 22 ththth Variance (Mean Square Deviation) of the Mean Field h(t)

In the Thermodynamic Limit of N→∞, > 0 for the Coherent State → 0 for the Incoherent State

∆a=0.32,σ=0.001

ε

ε*~0.0291

∆

εIncoherent State Coherent State

16

Scaling for the Mean Field

N

jj txf

Nth

1

))((1

)(Mean Field : Exhibiting the ‘2’-scaling (=-2.502 987 …)

Return Maps of the Mean Field for

a=0.32/i, =0.02/2i, =0.001/i, t =2i

[i (level of renormalization)=0, 1, 2]

Orbital Scalings in the Logistic Map

‘’-scaling near the Critical Point x=0

‘2’-scaling near the First Iterate of the Critical Point x=1 [= f(0)]

1)t(hi increases, level theAs

410N

21 1 tt axx

x=1 (most concentrated region)

x=0 (most rarified region)

1

2

1

) h(2i

h(2

τ+

1))

τ

17

Appearance of Similar Coherent States of Higher Orders (N=10 )

1st-Order Renormalized State (a=0.32/, =0.001/)

2nd-Order Renormalized State (a=0.32/2, =0.001/2)

3rd-Order Renormalized State (a=0.32/3, =0.001/3)

=0.039/23

=0.039/22

=0.039/2

Bifurcation Diagram Variance Diagram Return Map

4

18

Self-Similar State Diagrams (N=10 )

Incoherent States (White)Coherent States (Gray)

1st Renormalized State Diagram 2nd Renormalized State Diagram=0.001/ =0.001/2

=0.001

4

19

Effect of Noise on the Transition to Coherence

Scaling for the State Diagram in the - Plane

Multiple Transitions to Coherence for < * (=0.003)

Single Period-Doubling Transition to Coherence for > *

1st Order Renormalized State Diagram

a=0.32/

2nd Order Renormalized State Diagram

0th Order State Diagram

a=0.32/2

20

A Global Population of Inertially Coupled Maps

....,,1),...,,,()())(()1(11

Nixxxgttxftxiiiiii

Renormalization Result for the Scaling of the Coupling Parameter:

Nonlinear Coupling [Tendency of equalizing the states of the elements Dissipative coupling]

coupling quadratic :))(())((1

)...,,,( :e.g.1

11 txftxfN

xxxg i

N

jjiii

Linear Coupling

One Relevant Scaling Factor: =2

)()(1

)...,,,(1

11 txtxN

xxxg i

N

jjiii

Two Relevant Scaling Factors:

1 = (=-2.502 987 …): inertial coupling (each element: maintaining the memory of its previous states)

Combination of the Linear and Quadratic Coupling

Pure Inertial Coupling with Only One Relevant Scaling Factor: =

))](())((1

[088.0)]()(1

[)...,,,(11

11 txftxfN

txtxN

xxxg i

N

jji

N

jjiii

[Ref: S.P. Kuznetsov, Chaos, Solitons, and Fractals 2, 281-301 (1992).]

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

2 = 2

21

Scaling near the Zero-Coupling Critical Point (N=10 ) Universal Scaling near the Zero-Coupling Critical Point (a, , ) = (a*, 0, 0): (a* = 1.401 155 189 …)

Dynamical Behavior at a Set of Parameters (a, , ) Dynamical Behavior [with Doubled Time Scale (t=2)] at a Set of Renormalized Parameters (a/, / c, /) [=4.669201, =6.61903]

Asynchronous Chaotic Attractors (containing the diagonal)

“Scaling Factor for the Coupling Parameter for the Inertial Coupling:

/.

,/.,/.a

0030

020320

0030

020320

.

,.,.a

2

22

0030

020320

/.

,/.,/.a

[ 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).]

)5029872( ....c

4

22

Onset of Coherence

Bifurcation Diagram and Return Map of the Mean Field k(t) (N=10 )

N

j

T

tT

j tktktΔk,tkT

tk,txN

tk1

)()()()(1

lim)()(1

)(

)( FieldMean theof Variance:))()(())(( 22 tktktktk

Order Parameter

0340.*r

0070.*

l

Δa=0.32σ=0.003

Coherent Incoherent Coherent

4

Coherent State for ε=0.04Incoherent State for ε=0.02

23

Scaling for the Mean Field

N

jj tx

Ntk

1

)(1

)(Mean Field : Exhibiting the ‘’-scaling (=-2.502 987 …)

Return Maps of the Mean Field for

a=0.32/i, =0.02/2i, =0.003/i, t =2i

[i (level of renormalization)=0, 1, 2]

Orbital Scalings in the Logistic Map

‘’-scaling near the Critical Point x=0

0)(increases, level theAs tki

410N

21 1 tt axx

x=1 (most concentrated region)

x=0 (most rarified region)

1

2

1

24

Scaling Associated with the Mean Field k(t) (N=10 )

1st-Order Renormalized State (a=0.32, =0.003)

2nd-Order Renormalized State (a=0.32/, =0.003/)

3rd-Order Renormalized State (a=0.32/2, =0.003/2)

=0.04/2

=0.04/

=0.04

Bifurcation Diagram Variance Diagram Return Map

N

jj

txN

tk1

)(1

)( :FieldMean

4

25

Coherence and Scaling in a Heterogeneous Ensemble ofGlobally Coupled Maps

Heterogeneous Ensemble (consisting of non-identical elements)

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

)),(()1(1

NiatxfatxfN

atxftx ii

N

jjjiii

Spread in the map parameter ai for each element: Randomly chosen with uniform distribution in the interval of (a-, a+) (: spread parameter)

Transition from an Incoherent to a Coherent State

N

jj txf

Nth

1

))((1

)( :FieldMean

Occurrence of a Coherence at =* (=0.0289) for a=0.32 and =0.006 Coherent State for =0.03Incoherent State for ε=0.02

26

Return Map

Successive Appearance of Similar Coherent States of Higher Orders

1st-Order Renormalized State (a=0.32/, =0.006/)

2nd-Order Renormalized State (a=0.32/2, =0.006/2)

3rd-Order Renormalized State (a=0.32/3, =0.006/3)

Bifurcation Diagram Variance Diagram

=0.038/2

=0.038/22

=0.038/23

27

An Ensemble of Globally Coupled Noisy Pendulums Globally Coupled Noisy Pendulums

Phase Coherent Attractor in the Single Pendulum

.2sin)2cos(22),,(],1

[),,(, 2

1

xtAytyxfyyN

tyxfyyx i

N

jjiiiiii

strength. Noise :),()()(n correlatio delta and 0mean zero a with noise hiteGaussian w : 2121 tttt

)(

)(tan 1

tx

ty

22 yx

yxyx

...),,n(

nf||

21

2

at peaks

N

jj

N

jj

tyN

tYtxN

tX11

)(1

)(,)(1

)(A=0.06=0.0015=0.2

=2, =1A(A-A*)=0.06 (A*=6.57615)=0.0015

Phase Coherence (Flow) and Amplitude Incoherence

• Mean Field: Rotation on a Noisy Limit Cycle (Flow), Noisy Stationary State (Map)

28

Onset of Coherence

Bifurcation Diagram and Return Map of the Mean Field H(n) (N=10 )

)n(H))n(H)n(H())n(H( theof Variance:22

Order Parameter

33670.*r

Incoherent Coherent

4

Coherent State for ε=0.04Incoherent State for ε=0.02

N

jj )n(y

N)n(H

1

1 :FieldMean

29

1st-Order Renormalized State (A=0.06/, =0.0015/)

2nd-Order Renormalized State (A=0.06/2, =0.0015/2)

3rd-Order Renormalized State (A=0.06/3, =0.0015/3)

Scaling Associated with the Mean Field H(n) (N=10 )4

=0.3/2

=0.3/22

=0.3/23

Return MapMSD DiagramBifurcation Diagram

30

Summary

Investigation of Onset of Coherence in an Ensemble of Globally Coupled Noisy Logistic Maps

Universality for the Results

Confirmed in a Population of Globally Coupled Noisy Pendulums

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

Successive Appearance of Similar Coherent States of Higher Orders

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