Time delay in the Kuramoto model with bimodal frequency distribution

Abstract

We investigate the effects of a time-delayed all-to-all coupling scheme in a large population of oscillators with natural frequencies following a bimodal distribution. The regions of parameter space corresponding to synchronized and incoherent solutions are obtained both numerically and analytically for particular frequency distributions. In particular we find that bimodality introduces a new time scale that results in a quasiperiodic disposition of the regions of incoherence.The Kuramoto model [1] is presumably the most successful attempt to study macroscopic synchronization phenomena arising in large heterogeneous populations of interacting self-oscillatory units [2,3,4]. Kuramoto, motivated by Winfree's work on biological oscillators [5], showed that the dynamics of an ensemble of N weakly interacting limit-cycle oscillators can be treated considering simply the oscillator phases (θ 1 , . . . , θ N ). In this paper we study the Kuramoto model with delayed interactions [6] where heterogeneity is established considering a certain distribution g(ω) of the natural frequencies ω i . The terms ξ i (t) represent uncorrelated zero-mean white noise processes,In absence of time delay (τ = 0) and for large N , as the coupling strength K exceeds a critical threshold K c the model (1) shows drastically different transitions to collective synchronization, depending on the shape of g(ω) [1,2,3,7,8,9]. For a strictly unimodal distribution the transition occurs between a totally incoherent state and a partially synchronized state. In contrast, symmetric bimodal distributions give rise to hysteresis and/or a transition to a time dependent state composed of two clusters [8,9].The interactions in ensembles of coupled oscillators have been traditionally considered to be instantaneous, an assumption that considerably simplifies the analysis of such systems. However, the study of phase oscillators with time-delayed coupling [21] is receiving interest since a number of theoretical studies show that time delay may considerably affect the synchronization phenomena, typically leading to multistability of many synchronous states (see e.g. [3,4] and references therein). In par- * Present address: Instituto de Física de Cantabria (CSIC-UC), Santander, Spain. ticular, the Kuramoto model with unimodal frequency distribution has been generalized to allow time-delayed interactions in [6,10]. Additionally, phase models with time delay have successfully explained synchronization between plasmodial oscillators [11] and in semiconductor laser arrays [12]. Recent studies also demonstrate that time delay may be a useful synchronization-control mechanism in large oscillatory populations [13].In this paper we investigate the effects of a bimodal frequency distribution on the Kuramoto model (1) with time delay τ . For τ = 0 and assuming that g(ω) is symmetric (centered atω with twin peaks of width γ at both sides), one may always transform to a rotating frame, such that the model (1) is symmetric under the reflection: (θ i , ω i ) → (−θ i , −ω i ). Ti...

The Kuramoto model in complex networks

“…In networked delay-coupled oscillators, research has been focusing on, e.g., uniform time delay [179,183,184,187,[189][190][191][192][193][194], uniform time delay either with phase shift [195] or with noise [189] or both of them [183,188], and distancedependent time delays [196][197][198][199][200][201]. The time delay yields rich phenomena including bistability between synchronized and incoherent states, unsteady solutions with time-dependent order parameters [183,187], and multistabilities where synchronized states coexist with stable incoherent states [193], and it may result in suppression of the collective frequency [193].…”

“…Next, let us consider networks of oscillators with the same constant time delay for all interactions [183,189,191,[193][194][195]. Recall that time delays induce various solutions, e.g., bistability between synchronized and incoherent states, unsteady solutions with time-dependent order parameters [183,187], and multistabilities where synchronized states coexist with stable incoherent states [193].…”

“…For bimodal natural frequency distributions, Montbrió et al [189] found that bimodality induces a quasiperiodic pattern thanks to the existence of a new time scale. Moreover, the stability boundaries of the incoherent state have a completely different structure in the parameter space in contrast to the case with unimodal distribution [183].…”

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“…As a final remark on the distribution of the W i let us underline that the power source can [10], also in presence of a time delay [11]. The underdamped case has been considered in Ref.…”

Stationary and Traveling Wave States of the Kuramoto Model with an Arbitrary Distribution of Frequencies and Coupling Strengths

“…Many KM modifications have been considered, e.g. with: nonisochronicity [6][7][8]; frequency adaptation [9]; time-varying parameters [10]; higher order [11], timedelayed [12,13] and nonlocal [14] couplings; and different oscillator communities [15,16].…”