Aging transition and clustering in a large population of coupled oscillators

1. 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 of OPU

2. Contents 1. Background 2. Aging in globally coupled oscillators# Aging transition: Examples Universal scaling Clustering of active oscillators Summary 3. Aging in locally coupled oscillators System size dependence of pc Summary # in collaboration with Kenji Nakanishi.

3. Deterioration 1. Background The problem of aging HD&KN, PRL 93 (2004), 104101. Coupled oscillators often model biological or physiological systems. due to aging or accidents etc. damped oscillator self-sustained oscillator ( inactive oscillator) ( active oscillator )

4. effects of bad components What happens when the ratio of inactive elements increases ? ( “aging” ) active inactive How robust is the activity of coupled oscillators against aging ? important not only biologically but technologically

5. 2. Aging in globally coupled oscillators Globally and diffusively coupled oscillators periodic or chaotic General form 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.

6. Aging transition: examples (1) Coupled Stuart-Landau equations Active oscillators a Inactive oscillators -b

7. The behavior of an order parameter A measure of macroscopic activity vanishes at N=1000 ( Aging transition ) K=3, p=0.6 Synchronization within each group

8. Theory Reduction to a four-dimensional system q=1-p The death stabilized at death Aging transition

9. limit-cycle fixed point (2) Coupled Roessler systems N=100 Parameters (1)

10. Parameters (2) chaos Reverse period-doubling cascade

11. Universal Scaling at the aging transition Example Coupled periodic Roessler systems N=1000

12. General theory active Reduction inactive 1. The reduction is possible. 2. A fixed point exists. 3. Its becomes Hopf unstable at Assumptions

13. Hopf Universal crossover scaling

14. Clustering of active oscillators The synchronization within the active group breaks down in a region of the parameter plane. Resonance-like enhancement of inhomogeneity measured by the bar average over all j < > time average Diffusion-induced inhomogeneity below the peak point under the scalar type diffusive coupling: Coupled SL equations N=4000, p=0 Dx=Dy=K (x=Re(z), y=Im(z))

15. Analysis of the simplest 2-cluster state Example of the cluster structure cluster set of perfectly synchronized oscillators fractions of clusters in the order of the size from above except the largest 2-cluster states with one cluster much smaller than the other are abundant near the both ends of the clustering region

16. periodic oscillation Simplest 2-cluster state main cluster oscillators 1 to N-1 oscillator N 1 N-1 Approximation in the large system-size limit main cluster is unaffected by oscillator N oscillator N obeys perfect sync (stable) u=1

17. periodic aperiodic × Theory vs. simulation (N=1000) c2=-3,K=0.51, M=4 quasiperiodic periodic theoretical curves SN: saddle-node bifurcation middle: Hopf bifurcation SC: saddle connection c2=-3,K=0.94, M=9

18. 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 • Clustering and Swing-by mechanism of • Diffusion-induced inhomogeneity

19. 3. 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 2 N >> 1 1 1 N N-1 Aging proceeds through random inactivation of oscillators

20. 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 active oscillator →　limit-cycle For K=0 inactive oscillator 　　 → z=0

21. 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.

22. (K,p) phase diagram Example: a=b=1, c1=1, c2=-0.5 inactive phase zj=0 for all j p Aging transition boundaries N = 6400 ( 20) 1600 ( 40) 400 ( 50) 100 (100) active phase K Number of realizations Δp=0.01

23. 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…

24. (2) Vanishing of the inactive region for N→ ∞ Simulation results suggest pc(K,N) → 1 for N→ ∞ with K fixed 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

25. Scaling behavior of pc How does pc approach unity as N grows toward infinity ? Example a=b=1, c1=1, c2=-0.5 1-pc(N) vs. N N = 100 ～ 12800 K = 1, 1.8, 2.2, 2.6, 3.3, 4 Power laws !

26. K dependence of the power law exponent fit range N=100 ～ 12800 γ takes small values and tends to decrease with K.

27. 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

28. Main 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.