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Spontaneous synchronization of coupled oscillator systems with frequency adaptation

Dane Taylor, Edward Ott, and Juan G. Restrepo
Phys. Rev. E 81, 046214 – Published 27 April 2010
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  • INTRODUCTION
  • FREQUENCY ADAPTION MODEL
  • SPONTANEOUS STATE TRANSITIONS
  • DISCUSSION
  • CONCLUSIONS
  • ACKNOWLEDGEMENTS
    • References

    Abstract

    We study the synchronization of Kuramoto oscillators with all-to-all coupling in the presence of slow, noisy frequency adaptation. In this paper, we develop a model for oscillators, which adapt both their phases and frequencies. It is found that this model naturally reproduces some observed phenomena that are not qualitatively produced by the standard Kuramoto model, such as long waiting times before the synchronization of clapping audiences. By assuming a self-consistent steady state solution, we find three stability regimes for the coupling constant k, separated by critical points k1 and k2: (i) for k<k1 only the stable incoherent state exists; (ii) for k>k2, the incoherent state becomes unstable and only the synchronized state exists; and (iii) for k1<k<k2 both the incoherent and synchronized states are stable. In the bistable regime spontaneous transitions between the incoherent and synchronized states are observed for finite ensembles. These transitions are well described as a stochastic process on the order parameter r undergoing fluctuations due to the system’s finite size, leading to the following conclusions: (a) in the bistable regime, the average waiting time of an incoherent→coherent transition can be predicted by using Kramer’s escape time formula and grows exponentially with the number of oscillators; (b) when the incoherent state is unstable (k>k2), the average waiting time grows logarithmically with the number of oscillators.

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    • Received 19 January 2010

    DOI:https://doi.org/10.1103/PhysRevE.81.046214

    ©2010 American Physical Society

    Authors & Affiliations

    Dane Taylor1,*, Edward Ott2, and Juan G. Restrepo3

    • 1Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA
    • 2Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
    • 3Applied Mathematics Department, University of Colorado, Boulder, Colorado 80309, USA

    • *dane.taylor@colorado.edu

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    Vol. 81, Iss. 4 — April 2010

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    Images

    • Figure 1
      Figure 1
      Stable [upper black line, rs(k)] and unstable [lower gray lines, ru(k)] branches are shown for D=0.01, τ=50, and L=5,10,15,20 and the curve kr=2σ (dotted line), above which the frequency distribution is normalizable (note that rs(k)>2σ∀k,L). The values of k1 and k2 for L=5 are indicated at k1≈1.8 and k2≈6.37.Reuse & Permissions
    • Figure 2
      Figure 2
      (a) Order parameter r obtained from numerical simulation of Eqs. (1, 2) for decreasing values of k for N=104 (triangles) with D=0.01, τ=50, and L=5. The solid and dashed lines indicate stable rs(k) and unstable ru(k) solutions to Eq. (11), respectively. The letters a, b and c indicate values of k at which the frequency distribution is sampled for Fig. 2b. (b) Frequency distribution obtained directly from Eqs. (1, 2) (symbols) and from Eq. (10) (black lines). Curves labeled a, b, and c correspond to k=1, 2, and 4, indicated by arrows in Fig. 2a. The dashed vertical lines indicate ±kr for k=2.Reuse & Permissions
    • Figure 3
      Figure 3
      (Color online) For increasing coupling strength, synchronization occurs for each network when the order parameter fluctuations Δr allow r to surmount the barrier of the unstable solution ru(k) (dashed line). Simulation used D=0.01, τ=50, and L=5. Note that the transition coupling strengths k∗ approach k2 as network size N increases.Reuse & Permissions
    • Figure 4
      Figure 4
      State transitions parameterized by r for k1<k<k2 are schematically shown as Brownian motion in a one-dimensional energy landscape with two stable equilibria.Reuse & Permissions
    • Figure 5
      Figure 5
      Synchronization time τsync averaged over 100 realizations as a function of the number of oscillators N for k=6, which is within the bistable regime. (D=0.01, τ=50, and L=5)Reuse & Permissions
    • Figure 6
      Figure 6
      Synchronization time τsync averaged over 100 realizations as a function of the number of oscillators N for k=7>k2. (D=0.01, τ=50, and L=5). Note that the scale is different than that of Fig. 5.Reuse & Permissions
    • Figure 7
      Figure 7
      Spontaneous bidirectional transitions between the synchronized (dashed) and incoherent (dotted) states are observed for N=10, k=1.9, D=0.01, τ=50, and L=5. Note that because of the small system size, the incoherent state has an average order parameter of ⟨r⟩∼0.4.Reuse & Permissions
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