Aging transition and clustering in a large population of coupled oscillators

Aging transition and clustering in a large population of coupled oscillators

Hiroaki DAIDO

Department of Mathematical Sciences Graduate School of Engineering Osaka Prefecture University Sakai 599-8531, JAPAN

KIAS conference: NSPCS2008 (Seoul, July 2008)

emblem ofOPU

Contents

1. Background2. Aging in globally coupled oscillators#

Aging transition: Examples Universal scaling Clustering of active oscillators Summary3. Aging in locally coupled oscillators System size dependence of pc

Summary

# in collaboration with Kenji Nakanishi. 　

1. 1. BackgroundBackground

Coupled oscillators often model biological or physiological systems.

Deterioration due to aging or accidents etc.

self-sustained oscillator damped oscillator

( active oscillator ) ( inactive oscillator )

The problem of aging HD&KN, PRL 93 (2004), 104101.

What happens when the ratio of inactive elements increases ?

( “aging” )active

inactive

How robust is the activity of coupled oscillatorsagainst aging ?

important not only biologically but technologically

effects of bad components

Globally and diffusively coupled oscillators

periodic or chaoticGeneral form

2. Aging in globally coupled oscillators2. Aging in globally coupled oscillators

Examples (laboratory experiments)

Coupled electrochemical reaction systems Kiss et al. Science 296(2002), 176.

Coupled salt-water oscillators Miyakawa et al. Physica D 151(2001), 217.

(1) Coupled Stuart-Landau equations

Active oscillators

Inactive oscillators

Aging transition: examples

a

-b

The behavior of an order parameter

A measure of macroscopic activity

Synchronization withineach group

N=1000vanishes at

( Aging transition )

K=3, p=0.6

Theory

Reduction to a four-dimensional system

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

death

Aging transition

The death stabilized at

q=1-p

(2) Coupled Roessler systems

limit-cycle

fixed point

Parameters (1)

N=100

Parameters (2)

chaos

Reverse period-doubling cascade

Universal Scaling at the aging transition

ExampleCoupled periodic Roessler systemsN=1000

General theory

Assumptions

Reduction

1. The reduction is possible.

2. A fixed point exists.

3. Its becomes Hopf unstable at

active

inactive

Hopf

Universalcrossoverscaling

Clustering of active oscillators

The synchronization within the active group breaks down in a region of the parameter plane.

Coupled SL equations N=4000, p=0

Resonance-like enhancementof inhomogeneity measured by

the bar average over all j

Diffusion-induced inhomogeneitybelow the peak point under thescalar type diffusive coupling:

Dx=Dy=K (x=Re(z), y=Im(z))

< > time average

Analysis of the simplest 2-cluster state

Example of the cluster structure

clusterset of perfectlysynchronized oscillators

fractions of clusters in theorder of the size from aboveexcept the largest

2-cluster states with one cluster much smaller than the otherare abundant near the both ends of the clustering region

Approximation in the large system-size limit

Simplest 2-cluster state

oscillators 1 to N-1oscillator N

main cluster

N-1 1

main cluster is unaffected by oscillator N

periodic oscillation

oscillator N obeys

u=1 perfect sync (stable)

Theory vs. simulation

c2=-3,K=0.51, M=4

c2=-3,K=0.94, M=9

quasiperiodic

periodic

SN: saddle-node bifurcationmiddle: Hopf bifurcationSC: saddle connection

theoretical curves

periodicaperiodic×

(N=1000)

Summary of Part 1

１． The problem of aging : Effects of increasing 　　 inactive elements

２． Aging transition

Strong coupling: favorable for coherence, but less robust against aging !

３． Universal scaling at the aging transition

4. Clustering and Swing-by mechanism of

Diffusion-induced inhomogeneity

3. Aging in locally coupled oscillators3. Aging in locally coupled oscillators

Effects of aging in locally coupled oscillators

a chain under the periodic boundary condition

(i.e. a ring) as a first step

1

Aging proceeds through random inactivation of oscillators

N >> 1

19

1

2

N

N-1

Model & methods

Coupled Stuart-Landau oscillators on a ring

for all active oscillators (a>0)

for all inactive oscillators (b>0)

Number ratio: active: inactive=1-p:p

For K=0

active oscillator → limit-cycle

inactive oscillator　　 → z=0 20

Aging scheme

Randomly choosing some active sites to inactivate at each step of increasing p .

The chosen oscillators remain inactive for all p after this.

Then, results are averaged over many realizations.

21

(K,p) phase diagram

Example: a=b=1, c1=1, c2=-0.5

Aging transition boundaries

N = 6400 ( 20) 1600 ( 40) 400 ( 50) 100 (100)

active phase

inactive phase

Δp=0.0122

zj=0 for all j

Number of realizationsK

p

Key features of the phase diagram

(1) Existence of Kc insensitive to changes in N

A linear stability analysis of the inactive state for p=(N-1)/N shows that Kc is given, for N → ∞, by

Example

a=b=1, c1=1 Kc=0.648…23

(2) Vanishing of the inactive region for N → ∞

pc(K,N) → 1 for N → ∞ with K fixed

Simulation results suggest

Absence of the aging transition in the thermodynamic limit

Note: This does not imply unimportance of the AT, because (1) convergence of pc is slow, and (2) system sizes of real coupled oscillators are not always huge.

e.g. Lamprey’s CPG N ～ 100 mammalian circadian clocks N ～ 10000

24

Scaling behavior of pc

How does pc approach unity as N grows toward infinity ?

Example a=b=1, c1=1, c2=-0.5

N = 100 ～ 12800K = 1, 1.8, 2.2, 2.6, 3.3, 4

Power laws !

1-pc(N) vs. N

25

K dependence of the power law exponent

fit range N=100 ～ 12800

γ takes small valuesand tends to decreasewith K.

26

Summary of Part 2

Aging in locally coupled oscillators

a chain of Stuart-Landau oscillators with n. n. interactions (as a first step)

(1) Aging transition (AT) in finite-size systems

Existence of Kc

(2) Absence of AT in the thermodynamic limit

27

28

Main referencesMain references

H. D. & K. Nakanishi,

PRL 93(2004), 104101; 96(2006),

054101. PRE 75(2007),

056206; 76(2007), 056206(E).

H. D., to be published.