Phase synchronization on scale-free and random networks in the presence of noise

Khoshbakht, Hamid; Shahbazi, Farhad; Samani, Keivan Aghababaei

doi:10.1088/1742-5468/2008/10/p10020

Cited by 16 publications

(22 citation statements)

References 28 publications

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“…Besides the theoretical approaches described above, other aspects of the dynamics of the stochastic Kuramoto model were numerically addressed [283,[291][292][293][294][295]. In particular, Traxl et al [283] developed a numerical framework and systematically analyzed the influence of noise and coupling strength on the maximum degree of synchronization of several networks; including fully connected, random, modular and real topologies.…”

Section: Gaussian Approximationmentioning

confidence: 99%

The Kuramoto model in complex networks

et al. 2016

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Synchronization of an ensemble of oscillators is an emergent phenomenon present in several complex systems, ranging from social and physical to biological and technological systems. The most successful approach to describe how coherent behavior emerges in these complex systems is given by the paradigmatic Kuramoto model. This model has been traditionally studied in complete graphs. However, besides being intrinsically dynamical, complex systems present very heterogeneous structure, which can be represented as complex networks. This report is dedicated to review main contributions in the field of synchronization in networks of Kuramoto oscillators. In particular, we provide an overview of the impact of network patterns on the local and global dynamics of coupled phase oscillators. We cover many relevant topics, which encompass a description of the most used analytical approaches and the analysis of several numerical results. Furthermore, we discuss recent developments on variations of the Kuramoto model in networks, including the presence of noise and inertia. The rich potential for applications is discussed for special fields in engineering, neuroscience, physics and Earth science. Finally, we conclude by discussing problems that remain open after the last decade of intensive research on the Kuramoto model and point out some promising directions for future research.

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Section: Gaussian Approximationmentioning

confidence: 99%

The Kuramoto model in complex networks

et al. 2016

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“…In our numerical work, we choose a box distribution in the interval [−w/2, w/2] for η, so that its variance is equal to D = w 2 /24. It can be shown that by proper re-scaling of the time variable, the effect of parameters D and K can be included in a single parameter g 2 = D K [15], converting the dynamical equations to:…”

Section: The Effect Of Random Forcementioning

confidence: 99%

“…One of such models has been proposed by Kuramoto, which consists of a set of oscillators with fixed amplitude (phase oscillators) mutually coupled by a 2π periodic interaction [13]. The stochastic Kuramoto model has been studied on the globally connected [14] networks [15]. Analytical results on an all-to-all network show that for a given distribution of intrinsic frequencies of oscillators, a minimum value of coupling is needed for synchronization.…”

Section: Introductionmentioning

confidence: 99%

Noise-induced synchronization in small world networks of phase oscillators

2012

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A small-world (SW) network of similar phase oscillators, interacting according to the Kuramoto model, is studied numerically. It is shown that deterministic Kuramoto dynamics on SW networks has various stable stationary states. This can be attributed to the so-called defect patterns in an SW network, which it inherits from deformation of helical patterns in its regular parent. Turning on an uncorrelated random force causes vanishing of the defect patterns, hence increasing the synchronization among oscillators for moderate noise intensities. This phenomenon, called stochastic synchronization, is generally observed in some natural networks such as the brain neural network.

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“…, P t (τ )) T , which evolves according to P t+1 = F ( P t ). Note that F 1 is the only component of F which is nonlinear [see (13)]. From (9), the fixed point for any 2 ≤ s ≤ τ −1 is clearly P * s ≡ P ∞ (s) = P ∞ (s−1) = · · · = P * 1 which, upon substitution in (10), gives P * τ = P * 1 /p γ .…”

Section: Mean-field Analysismentioning

confidence: 99%

Collective oscillations of excitable elements: order parameters, bistability and the role of stochasticity

Rozenblit¹,

Copelli²

2011

J. Stat. Mech.

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We study the effects of a probabilistic refractory period in the collective behavior of coupled discrete-time excitable cells (SIRS-like cellular automata). Using mean-field analysis and simulations, we show that a synchronized phase with stable collective oscillations exists even with non-deterministic refractory periods. Moreover, further increasing the coupling strength leads to a reentrant transition, where the synchronized phase loses stability. In an intermediate regime, we also observe bistability (and consequently hysteresis) between a synchronized phase and an active but incoherent phase without oscillations. The onset of the oscillations appears in the mean-field equations as a Neimark-Sacker bifurcation, the nature of which (i.e. super-or subcritical) is determined by the first Lyapunov coefficient. This allows us to determine the borders of the oscillating and of the bistable regions. The mean-field prediction thus obtained agrees quantitatively with simulations of complete graphs and, for random graphs, qualitatively predicts the overall structure of the phase diagram. The latter can be obtained from simulations by defining an order parameter q suited for detecting collective oscillations of excitable elements. We briefly review other commonly used order parameters and show (via data collapse) that q satisfies the expected finite size scaling relations.

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Phase synchronization on scale-free and random networks in the presence of noise

Cited by 16 publications

References 28 publications

The Kuramoto model in complex networks

The Kuramoto model in complex networks

Noise-induced synchronization in small world networks of phase oscillators

Collective oscillations of excitable elements: order parameters, bistability and the role of stochasticity

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