Thermodynamic limit of the first-order phase transition in the Kuramoto model

Pazó, Diego

doi:10.1103/physreve.72.046211

Cited by 201 publications

(221 citation statements)

References 22 publications

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“…5(f), as the coupling strength K is varied, the order parameter R of the phase oscillators interconnected by the minimal ES network exhibits a sudden hysteretic change, associated to a discontinuous phase transition, whereas the system with a unimodal frequency distribution undergoes a continuous phase transition [5]. This discontinuous phase transition, also known as "explosive synchronisation", has been studied in the literature [18][19][20][21], also in the case of adaptive networks [22,23], revealing that the correlation between natural frequencies and the node degree, as shown in Fig. 4(d), can induce this phenomenon.…”

Section: Emergence Of Minimal Networkmentioning

confidence: 99%

Heterogeneity induces emergent functional networks for synchronization

2015

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We study the evolution of heterogeneous networks of oscillators subject to a state-dependent interconnection rule. We find that heterogeneity in the node dynamics is key in organizing the architecture of the functional emerging networks. We demonstrate that increasing heterogeneity among the nodes in state-dependent networks of phase oscillators causes a differentiation in the activation probabilities of the links. This, in turn, yields the formation of hubs associated to nodes with larger distances from the average frequency of the ensemble. Our generic local evolutionary strategy can be used to solve a wide range of synchronization and control problems.

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Section: Emergence Of Minimal Networkmentioning

confidence: 99%

Heterogeneity induces emergent functional networks for synchronization

2015

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“…Such explosive synchronization is characterized by the emergence of a range of coupling strengths where incoherent and synchronized states are both stable, unlike the first-order transition studied in Ref. [16]. Subsequently, significant attention has been paid to the further exploration of degree-frequency correlations [17][18][19][20] and in particular explosive synchronization [21][22][23][24][25][26][27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Disorder induces explosive synchronization

Skardal

Arenas

2014

Phys. Rev. E

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We study explosive synchronization, a phenomenon characterized by first-order phase transitions between incoherent and synchronized states in networks of coupled oscillators. While explosive synchronization has been the subject of many recent studies, in each case strong conditions on either the heterogeneity of the network, its link weights, or its initial construction are imposed to engineer a first-order phase transition. This raises the question of how robust explosive synchronization is in view of more realistic structural and dynamical properties. Here we show that explosive synchronization can be induced in mildly heterogeneous networks by the addition of quenched disorder to the oscillators' frequencies, demonstrating that it is not only robust to, but moreover promoted by, this natural mechanism. We support these findings with numerical and analytical results, presenting simulations of a real neural network as well as a self-consistency theory used to study synthetic networks.

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“…The terms ξ i (t) represent uncorrelated zero-mean white noise processes, ξ i (t) = 0, ξ i (t)ξ j (t ′ ) = 2Dδ ij δ(t − t ′ ). 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.…”

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confidence: 99%

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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Thermodynamic limit of the first-order phase transition in the Kuramoto model

Cited by 201 publications

References 22 publications

Heterogeneity induces emergent functional networks for synchronization

Heterogeneity induces emergent functional networks for synchronization

Disorder induces explosive synchronization

Time delay in the Kuramoto model with bimodal frequency distribution

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