Stability of incoherence in a population of coupled oscillators

Strogatz, Steven H.; Mirollo, Renato

doi:10.1007/bf01029202

Cited by 496 publications

(452 citation statements)

References 15 publications

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“…In this context, the question of how the random frequencies influence synchronization has been raised by many authors, not only in the Kuramoto model ( [43]) but also for more general models of weakly interacting diffusions (e.g. neuronal models, see [4] and references therein).…”

Section: Synchronization Of Heterogeneous Oscillatorsmentioning

confidence: 99%

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Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Luçon¹

2014

Journal of Functional Analysis

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We investigate the long-time asymptotics of the fluctuation SPDE in the Kuramoto synchronization model. We establish the linear behavior for large time and weak disorder of the quenched limit fluctuations of the empirical measure of the particles around its McKean-Vlasov limit. This is carried out through a spectral analysis of the underlying unbounded evolution operator, using arguments of perturbation of self-adjoint operators and analytic semigroups. We state in particular a Jordan decomposition of the evolution operator which is the key point in order to show that the fluctuations of the disordered Kuramoto model are not self-averaging.

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Section: Synchronization Of Heterogeneous Oscillatorsmentioning

confidence: 99%

“…or in physical contexts (see [28,43] and references therein). While a precise description of each of the different instances in which synchronization emerges demands specific, possibly very complex models, the Kuramoto model [1] has emerged as capturing some of the fundamental aspects of synchronization.…”

Section: Synchronization Of Heterogeneous Oscillatorsmentioning

confidence: 99%

Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Luçon¹

2014

Journal of Functional Analysis

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“…[14] Strogatz and Mirollo provide a linear stability analysis of ρ 0 focussing especially on populations with even distributions (18) that decrease monotonically away from ω = 0. [7] They show that L has a continuous spectrum with real part equal to −D, and that it may also have point spectrum (eigenvalues) depending on the coupling strength. For K sufficiently small, the point spectrum is empty, but as the coupling increases a real eigenvalue λ emerges from the continuous spectrum and for K > K c moves into the right half plane λ > 0 signifying linear instability of ρ 0 .…”

Section: Introductionmentioning

confidence: 99%

“…However the theoretical explanation of this stability has proved to be rather subtle even for the incoherent state, but there has been significant recent progress in the work of Strogatz and Mirollo. [7,8] Following Sakaguchi, they considered the large N limit of (1) and studied the Fokker-Planck equation…”

Section: Introductionmentioning

confidence: 99%

Amplitude expansions for instabilities in populations of globally-coupled oscillators

1994

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We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite in contrast to the singular behavior found in similar instabilities described by the Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable travelling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and Okuda and Kuramoto predicted stable travelling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable travelling waves results from a failure to include all unstable modes.

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“…The main result that we present addresses the important issue of the stability of the non-trivial stationary profilesq(·), more precisely of the stability of the invariant manifold {q(· + θ 0 )} θ 0 ∈S . In the literature we find a full analysis of incoherence stability [22] (also in presence of disorder) as well as an analysis of synchronized profiles as bifurcation from the incoherent 1/2π profile (we refer to [1] and the several references therein). Our aim is to have a detailed non-perturbative analysis of the linearized evolution operator in the non disordered case, for every K > K c = 1.…”

Section: 4mentioning

confidence: 99%

Dynamical Aspects of Mean Field Plane Rotators and the Kuramoto Model

2009

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Abstract. The Kuramoto model has been introduced in order to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc... We look at the Kuramoto model with white noise forces: in mathematical terms it is a set of N oscillators, each driven by an independent Brownian motion with a constant drift, that is each oscillator has its own frequency, which, in general, changes from one oscillator to another (these frequencies are usually taken to be random and they may be viewed as a quenched disorder). The interactions between oscillators are of long range type (mean field). We review some results on the Kuramoto model from a statistical mechanics standpoint: we give in particular necessary and sufficient conditions for reversibility and we point out a formal analogy, in the N → ∞ limit, with local mean field models with conservative dynamics (an analogy that is exploited to identify in particular a Lyapunov functional in the reversible set-up). We then focus on the reversible Kuramoto model with sinusoidal interactions in the N → ∞ limit and analyze the stability of the non-trivial stationary profiles arising when the interaction parameter K is larger than its critical value Kc. We provide an analysis of the linear operator describing the time evolution in a neighborhood of the synchronized profile: we exhibit a Hilbert space in which this operator has a self-adjoint extension and we establish, as our main result, a spectral gap inequality for every K > Kc.

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Stability of incoherence in a population of coupled oscillators

Cited by 496 publications

References 15 publications

Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model

Amplitude expansions for instabilities in populations of globally-coupled oscillators

Dynamical Aspects of Mean Field Plane Rotators and the Kuramoto Model

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