Bifurcation analysis on the globally coupled Kuramoto oscillators with distributed time delays

Niu, Ben; Guo, Yuxiao

doi:10.1016/j.physd.2013.10.003

Cited by 15 publications

(20 citation statements)

References 24 publications

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“…In Section 4, the stability and direction of periodic solutions bifurcating from Hopf bifurcations are investigated by using the normal form theory and the center manifold theorem due to [31], which gives clearly the location where coherent states appear and their stability. In Section 5, inspired by the method given in [19,24,25], numerical simulations are carried out to support the obtained results.…”

Section: Introductionmentioning

confidence: 86%

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Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

2018

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In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained resultswhere several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed. * niubenhit@163.com

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Section: Introductionmentioning

confidence: 86%

“…show are about the bifurcation branches of the model, because bifurcation branches are widely used in literatures [19,24,25], which are also a quite unambiguous way to show the change of numbers of steady states or periodic oscillations.…”

Section: Numerical Examplesmentioning

confidence: 99%

Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

2018

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“…In addition, the distribution of natural frequencies, which determines the intrinsic property and classification of uncoupled oscillators, can influence the dynamics of coupled oscillators. There have been a series of works reporting on how the topological properties of the symmetric unimodal and bimodal distribution with or without time delays produce synchronization and even new dynamics [36][37][38][39][40][41][42][43][44][45]. However, in the literature, there have been only a few investigations on how the asymmetry, as well as the shift distance of the bimodal distribution of the natural frequencies, influences the elimination of synchronization in terms of the adaptive scheme with a feedback delay.…”

Section: Introductionmentioning

confidence: 99%

Adaptive elimination of synchronization in coupled oscillator

Zhou

et al. 2017

New J. Phys.

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We present here an adaptive control scheme with a feedback delay to achieve elimination of synchronization in a large population of coupled and synchronized oscillators. We validate the feasibility of this scheme not only in the coupled Kuramoto's oscillators with a unimodal or bimodal distribution of natural frequency, but also in two representative models of neuronal networks, namely, the FitzHugh-Nagumo spiking oscillators and the Hindmarsh-Rose bursting oscillators. More significantly, we analytically illustrate the feasibility of the proposed scheme with a feedback delay and reveal how the exact topological form of the bimodal natural frequency distribution influences the scheme performance. We anticipate that our developed scheme will deepen the understanding and refinement of those controllers, e.g. techniques of deep brain stimulation, which have been implemented in remedying some synchronization-induced mental disorders including Parkinson disease and epilepsy.

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“…Here, when a hysteresis loop is mentioned, we mean that coherent states and incoherent states coexist in the Kuramoto model when the parameter k is less than the Hopf bifurcation value. In [20], the authors have interpreted the appearance of subcritical Hopf bifurcations in the way of normal form analysis. However, a clear boundary between the supercritical and subcritical bifurcations (a degenerated case) has not been theoretically studied yet.…”

Section: Introductionmentioning

confidence: 99%

Codimension-two bifurcations induce hysteresis behavior and multistabilities in delay-coupled Kuramoto oscillators

Niu

2016

Nonlinear Dyn

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Hysteresis phenomena and multistability play crucial roles in the dynamics of coupled oscillators, which are now interpreted from the point of view of codimension-two bifurcations. On the Ott-Antonsen's manifold, complete bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling strength and delay as bifurcation parameters. It is rigorously proved that the system must undergo Bautin bifurcations for some critical values, thus there always exists saddlenode bifurcation of periodic solutions inducing hysteresis loop. With the aid of center manifold reduction method and the Matlab Package DDE-Biftool, the location of Bautin and double Hopf points and detailed dynamics are theoretically determined. We find that, near these critical points, at most four coherent states (two of which are stable) and a stable incoherent state may coexist, and that the system undergoes Neimark-Sacker bifurcation of periodic solutions. Finally, the clear scenarios about the synchronous transition in delayed Kuramoto model are depicted.

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Bifurcation analysis on the globally coupled Kuramoto oscillators with distributed time delays

Cited by 15 publications

References 24 publications

Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

Adaptive elimination of synchronization in coupled oscillator

Codimension-two bifurcations induce hysteresis behavior and multistabilities in delay-coupled Kuramoto oscillators

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