Intrinsic fluctuations and a phase transition in a class of large populations of interacting oscillators

The incoherent state of the Kuramoto model of coupled oscillators exhibits marginal modes in mean field theory. We demonstrate that corrections due to finite size effects render these modes stable in the subcritical case, i.e. when the population is not synchronous. This demonstration is facilitated by the construction of a non-equilibrium statistical field theoretic formulation of a generic model of coupled oscillators. This theory is consistent with previous results. In the all-to-all case, the fluctuations in this theory are due completely to finite size corrections, which can be calculated in an expansion in 1/N , where N is the number of oscillators. The N → ∞ limit of this theory is what is traditionally called mean field theory for the Kuramoto model.

Section: Field Theory For the Kuramoto Modelmentioning

confidence: 99%

Section: Field Theory For the Kuramoto Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Correlations, fluctuations, and stability of a finite-size network of coupled oscillators

Buice

Chow

2007

“…The discontinuous change in the amplitude is in a sharp contrast to a super-critical Hopf bifurcation at a collective level, where the amplitude of oscillation changes continuously at the bifurcation [19]. It should be noted that in a manner similar to that of the critical phenomenon in equilibrium systems, the continuous transition leads to a critical divergence of amplitude fluctuation [20]. (See also Refs.…”

mentioning

confidence: 99%

Critical phenomena in globally coupled excitable elements

Ohta

Sasa

2008

Critical phenomena in globally coupled excitable elements are studied by focusing on a saddlenode bifurcation at the collective level. Critical exponents that characterize divergent fluctuations of interspike intervals near the bifurcation are calculated theoretically. The calculated values appear to be in good agreement with those determined by numerical experiments. The relevance of our results to jamming transitions is also mentioned.PACS numbers: 82.40. Bj, 05.10.Gg, 89.75.Da,The understanding of cooperative phenomena in nonequilibrium systems is one of the most important problems in physics. In contrast to those in equilibrium systems, their nature essentially depends on the dynamical properties of the systems. This makes it difficult to perform a systematic study of the problem. Thus, it is necessary to investigate the typical cooperative phenomena in nonequilibrium systems.Recently, critical behaviors have been observed experimentally [1,2,3,4,5,6,7] and numerically [8,9,10,11,12,13,14,15] in typical examples of coupled excitable elements such as neural networks and cardiac tissues. In general, such critical behaviors are classified into several groups on the basis of the exponents of divergences. The classification enables the realization of a universality class for critical phenomena in coupled excitable elements. However, the broad distribution of the phenomena makes it difficult to elucidate the mechanism of the criticality. When we consider the role of the mean-field Ising model in theories of equilibrium statistical mechanics, it becomes apparent that it is necessary to develop an elegant method for theoretical analysis of a minimal model describing critical phenomena in coupled excitable elements.Thus, with this aim in mind, we analyze a previously proposed simple model for globally coupled excitable elements in this Letter [16]. In particular, we focus on a divergent behavior with respect to parameter change around a saddle-node bifurcation because the excitability of this model is related to the bifurcation. It should be noted that such transition properties have been studied for different excitable systems [2,8,10,11,14]. The main achievement of this Letter is a theoretical derivation of the critical exponents that characterize the singular behavior near a saddle-node bifurcation.Model: The excitable nature of a system is characterized by the existence of spikes in a time series. Mathematically speaking, spikes are described by trajectories near a homoclinic orbit in a differential equation. As a simple example, let us consider an ordinary differential equation ∂ t φ = ω−h sin φ for a phase variable φ ∈ [0, 2π], in which there exists the homoclinic orbit when ω = h. Then, when h is slightly larger than ω, a small perturbation for the fixed point φ * = sin −1 (ω/h) yields one spike. On the other hand, when h is slightly less than ω, the system shows an array of spikes with a long interspike interval. The qualitative change in the trajectories is an example of saddle-node bifurcation. By using thi...

“…For finite N , however, much less is known. The most important advances have come in three areas: finite-size corrections to the critical coupling at the phase transition, and finite-size scaling laws for the dynamical fluctuations of the order parameter just past the transition [11][12][13][14][15][16]; finite-N corrections to the model's kinetic theory [17,18]; and analysis of the model's phase-locked states and their bifurcations [19][20][21][22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Kuramoto model with uniformly spaced frequencies: Finite-Nasymptotics of the locking threshold

Ottino-Loffler

Strogatz

2016

We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, N , is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case, stable phase-locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width, called the locking threshold. For infinite N , the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite N have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when N 1. The leading correction to the infinite-N result scales like either N −3/2 or N −1 , depending on whether the frequencies are evenly spaced according to a midpoint rule or an endpoint rule. These scaling laws agree with numerical results obtained by Pazó [Phys. Rev. E 72, 046211 (2005)]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics. INTRODUCTIONIn 1975, Kuramoto proposed an elegant model of coupled nonlinear oscillators, now known as the Kuramoto model [1,2]. Since then the model has been applied to a wide range of physical, biological, chemical, social, and technological systems, and its analysis has stimulated theoretical work in nonlinear dynamics, statistical physics, network science, control theory, and pure mathematics. For reviews, see Refs. [3][4][5][6][7][8][9].The governing equations for the Kuramoto model arėfor i = 1, . . . , N , where θ i is the phase of oscillator i and ω i is its natural frequency. Inspired by Winfree's work on self-synchronizing systems of biological oscillators [10], Kuramoto restricted attention to attractive coupling, K > 0, and assumed that the ω j were randomly distributed across the population according to a prescribed probability distribution g(ω), which he took to be unimodal and symmetric about its mean. Without loss of generality the mean frequency can be set to 0 by going into a rotating frame, and the coupling K can be set to K = 1 by rescaling time. Both of these normalizations will be assumed in what follows. One adjustable parameter remains: the characteristic width γ of the frequency distribution. When γ is sufficiently large, numerical simulations show that the oscillators behave incoherently and run at their natural frequencies. At the other extreme, when γ = 0 the oscillators have identical frequencies and approach a perfectly synchronized solution with θ i (t) = θ j (t) for all i, j and t. Kuramoto's achievement was to analyze the dynamics of the model in between these two extremes. He solved the model in the continuum limit N → ∞ and obtained a number of beautiful results [1,2], opening up a fruitful line of research for many subsequent studies (reviewed in [3][4][5][6]). In particular, he showed that as γ is decreased from large values, a bifurcation takes place at a critical value γ c . This bifurcation gives rise to a branch of partially synchronized states: the ...