Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: bifurcation of the order function

Daido, Hiroaki

doi:10.1016/0167-2789(95)00260-x

Cited by 169 publications

(155 citation statements)

References 39 publications

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“…Then, using the machinery of Eqs. (14), (15), and (16), it is a straightforward but tedious exercise to numerically evaluate the first Lyapunov coefficient l 1 (µ) corresponding to the Hopf bifurcation occurring at (µ, a(µ)). As shown in Fig.…”

Section: B Continuous and Discontinuous Transitions In A Dichotomousmentioning

confidence: 99%

“…In addition, we study dichotomously disordered populations of oscillators and show that the bifurcation can be either supercritical or subcritical depending on the degree of disorder in the population. Such behaviors are reminiscent of a number of significantly more complex oscillator models [8,9,10,11,12,13,14], including Daido's generalized Kuramoto oscillators [15], where either disoder or microscopic coupling specifics can alter the nature (continuous or discontinuous) of the transition. However, our model provides a far simpler setting for observing both continous and discontinous transitions to synchronization.…”

Section: Introductionmentioning

confidence: 99%

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Continuous and discontinuous phase transitions and partial synchronization in stochastic three-state oscillators

et al. 2007

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We investigate both continuous (second-order) and discontinuous (first-order) transitions to macroscopic synchronization within a single class of discrete, stochastic (globally) phase-coupled oscillators. We provide analytical and numerical evidence that the continuity of the transition depends on the coupling coefficients and, in some nonuniform populations, on the degree of quenched disorder. Hence, in a relatively simple setting this class of models exhibits the qualitative behaviors characteristic of a variety of considerably more complicated models. In addition, we study the microscopic basis of synchronization above threshold and detail the counterintuitive subtleties relating measurements of time averaged frequencies and mean field oscillations. Most notably, we observe a state of suprathreshold partial synchronization in which time-averaged frequency measurements from individual oscillators do not correspond to the frequency of macroscopic oscillations observed in the population.

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Section: B Continuous and Discontinuous Transitions In A Dichotomousmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Continuous and discontinuous phase transitions and partial synchronization in stochastic three-state oscillators

et al. 2007

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“…For the analysis that follows, it is convenient to use the generalized order parameters [32] Z m ðtÞ ¼…”

Section: Low-dimensional Dynamics Of the Winfree Modelmentioning

confidence: 99%

Taking the Pulse

Laing¹

2014

Physics

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Large communities of biological oscillators show a prevalent tendency to self-organize in time. This cooperative phenomenon inspired Winfree to formulate a mathematical model that originated the theory of macroscopic synchronization. Despite its fundamental importance, a complete mathematical analysis of the model proposed by Winfree-consisting of a large population of all-to-all pulse-coupled oscillators-is still missing. Here, we show that the dynamics of the Winfree model evolves into the so-called OttAntonsen manifold. This important property allows for an exact description of this high-dimensional system in terms of a few macroscopic variables, and also allows for the full investigation of its dynamics. We find that brief pulses are capable of synchronizing heterogeneous ensembles that fail to synchronize with broad pulses, especially for certain phase-response curves. Finally, to further illustrate the potential of our results, we investigate the possibility of "chimera" states in populations of identical pulse-coupled oscillators. Chimeras are self-organized states in which the symmetry of a population is broken into a synchronous and an asynchronous part. Here, we derive three ordinary differential equations describing two coupled populations and uncover a variety of chimera states, including a new class with chaotic dynamics.

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“…(2). In the systems of globally coupled phase oscillators, it is known that such higher harmonic components in the coupling function are responsible for a rich variety of synchronous patterns excluded by the sine coupling [29][30][31][32][33][34]. Therefore we expect that also in the systems of nonlocally coupled phase oscillators with Eq.…”

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confidence: 99%

Persistent chimera states in nonlocally coupled phase oscillators

Suda

Okuda

2015

Phys. Rev. E

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Chimera states in the systems of nonlocally coupled phase oscillators are considered stable in the continuous limit of spatially distributed oscillators. However, it is reported that in the numerical simulations without taking such limit, chimera states are chaotic transient and finally collapse into the completely synchronous solution. In this Rapid Communication, we numerically study chimera states by using the coupling function different from the previous studies and obtain the result that chimera states can be stable even without taking the continuous limit, which we call the persistent chimera state. The behavior of coupled oscillator systems can describe various pattern formations in a wide range of scientific fields [1,2]. In the systems of nonlocally coupled identical oscillators, there often appears a strange phenomenon called the chimera state, which is characterized by the coexistence of coherent and incoherent domains, where the former domain consists of phase-locked oscillators and the latter domain consists of drifting oscillators with spatially changing frequencies . This interesting phenomenon was first discovered in the system of nonlocally coupled phase oscillators obeying the evolution equationwith 2π -periodic phases θ (x) on a finite interval x ∈ [0,1] under the periodic boundary condition, a smooth 2π -periodic coupling function , and the kernel G(y) = (κ/2) exp(−κ|y|), where a constant 1/κ denotes the coupling range [3]. Recently, similar spatiotemporal patterns have been found in various systems using, e.g., the logistic maps [14,15], Rössler systems [15], and FitzHugh-Nagumo oscillators [18].In the study of the chimera state, the system, Eq. (1), with the sine coupling [27]is particularly important because of its simplicity and generality. In fact, this coupling function was used also in the first discovery of the chimera state [3]. For numerical simulations, we usually discretize Eq. (1) into such form as Eq. (3). In the simulations of such discretized systems, we can confirm that chimera states are surely stable in the continuous limit N → ∞. However, the stability of chimera states in finitely discretized systems is questioned. In fact, it is reported that when N is finite, chimera states with the sine coupling are chaotic transient and finally collapse into the completely synchronous solution [12,13,26]. Recently, Ashwin and Burylko proposed the weak chimera similar to the chimera state, which is defined by the coexistence of frequency-synchronous and -asynchronous oscillators in the systems of coupled indistinguishable phase oscillators but is not necessarily spatially structured as coherent and incoherent domains [28]. They studied the weak chimera in some types of networks composed of the minimal number of oscillators with the Hansel-Mato-Meunier coupling, Eq. (4), and demonstrated that the weak chimera can be persistent (nontransient). In this Rapid Communication, we study chimera states in the systems of nonlocally coupled phase oscillators with the Hansel-MatoMeunier coupling b...

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Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: bifurcation of the order function

Cited by 169 publications

References 39 publications

Continuous and discontinuous phase transitions and partial synchronization in stochastic three-state oscillators

Continuous and discontinuous phase transitions and partial synchronization in stochastic three-state oscillators

Taking the Pulse

Persistent chimera states in nonlocally coupled phase oscillators

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