Synchronization of two interacting populations of oscillators

Montbrió, Ernest; Kurths, Jürgen; Blasius, Bernd

doi:10.1103/physreve.70.056125

Cited by 198 publications

(199 citation statements)

References 25 publications

(88 reference statements)

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“…Thus, at τ = τ n we expect to recover the typical reflection-symmetric stationary wave solutions found in the Kuramoto model with bimodal frequency distribution but without delay [9]. For general τ values the breaking of the reflection symmetry should give rise to the structures already observed for models without time delay but with either asymmetric bimodal frequency distributions [14] or asymmetric coupling functions [15].We begin our analysis by considering the noise-free case, D = 0, and a frequency distribution that consists of two infinitely sharp peaks (i.e. γ = 0):…”

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confidence: 91%

Time delay in the Kuramoto model with bimodal frequency distribution

2006

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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...

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confidence: 91%

Time delay in the Kuramoto model with bimodal frequency distribution

2006

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“…3B). Oscillating order parameters can arise in modified versions of the Kuramoto model, for example, in the presence of structure in the coupling topology among the oscillators (29). But here, the oscillations of the order parameter after perturbation reflect the periodic changes in the shape of the phase density, periodically shifting between unimodal and bimodal distributions.…”

Section: Mathematical Model Predictions and Agreement With Experimentmentioning

confidence: 99%

Cycles, phase synchronization, and entrainment in single-species phytoplankton populations

Massie

Blasius

Weithoff

et al. 2010

Proc. Natl. Acad. Sci. U.S.A.

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Complex dynamics, such as population cycles, can arise when the individual members of a population become synchronized. However, it is an open question how readily and through which mechanisms synchronization-driven cycles can occur in unstructured microbial populations. In experimental chemostats we studied large populations (>10 9 cells) of unicellular phytoplankton that displayed regular, inducible and reproducible population oscillations. Measurements of cell size distributions revealed that progression through the mitotic cycle was synchronized with the population cycles. A mathematical model that accounts for both the cell cycle and population-level processes suggests that cycles occur because individual cells become synchronized by interacting with one another through their common nutrient pool. An external perturbation by direct manipulation of the nutrient availability resulted in phase resetting, unmasking intrinsic oscillations and producing a transient collective cycle as the individuals gradually drift apart. Our study indicates a strong connection between complex within-cell processes and population dynamics, where synchronized cell cycles of unicellular phytoplankton provide sufficient population structure to cause small-amplitude oscillations at the population level.

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“…denotes the complex conjugate of the previous term. Substituting (5) into (3) and (2), one can derive an infinite set of integro-differential equations for the h n [15]. However, Ott and Antonsen [17] noticed that for the special choice…”

Section: Two Coupled Networkmentioning

confidence: 99%

“…As is well-known [15,17,9,14], if g k is a Lorentzian distribution the integral (8) can be evaluated analytically. Suppose that…”

Section: Two Coupled Networkmentioning

confidence: 99%

Chimera states in heterogeneous networks

Laing¹

2009

Chaos: An Interdisciplinary Journal of Nonlinear Science

212

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Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronise with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but here we study these states in heterogeneous models for which the natural frequencies of the oscillators are chosen from a distribution. For a model consisting of two subnetworks we obtain exact results by reduction to a finite set of differential equations, and for a network of oscillators in a ring we generalise known results. We find that heterogeneity can destroy chimerae, destroy all states except chimerae, or destabilise chimerae in Hopf bifurcations, depending on the form of the heterogeneity.Synchronisation of interacting oscillators is a problem of fundamental importance, with applications from Josephson junction circuits to neuroscience [19,24,22,27]. Since oscillators are unlikely to be identical, the effects of heterogeneity on their collective behaviour is of interest. One well-studied system of heterogeneous phase oscillators is the Kuramoto model [4, 9, 23], for which there is global coupling. Generalisations of this model with nonlocal coupling [3, 21, 2, 10, 16], or several populations of oscillators [1], have shown interesting types of behaviour referred to as "chimera" states in which some oscillators are synchronised with one another while the remainder are incoherent. Even though the effects of heterogeneity on synchronisation have been emphasised in the past [4,13,9,23], all chimerae have so far been studied in networks of identical oscillators. This raises the obvious question: do chimerae exist in networks of nonidentical oscillators? Here we address the question analytically, first using recent results to exactly derive a finite set of differential equations governing the dynamics of chimerae in two coupled networks of heterogeneous phase oscillators, and then using a similar idea to extend the results of Abrams and Strogatz [3] for chimera states in a ring of coupled oscillators.

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Synchronization of two interacting populations of oscillators

Cited by 198 publications

References 25 publications

Time delay in the Kuramoto model with bimodal frequency distribution

Time delay in the Kuramoto model with bimodal frequency distribution

Cycles, phase synchronization, and entrainment in single-species phytoplankton populations

Chimera states in heterogeneous networks

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