An infinite-period phase transition versus nucleation in a stochastic model of collective oscillations

Assis, Vladimir R. V.; Copelli, Mauro; Dickman, Ronald

doi:10.1088/1742-5468/2011/09/p09023

Cited by 13 publications

(29 citation statements)

References 27 publications

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“…For large values of a, the shape of the limit cycle becomes less circular, approaching the borders of the triangle, and global oscillations become highly anharmonic, with a finite fraction of the oscillators collectively spending a long time in each of the three states before "jumping" to the next state at a much shorter time scale. At a = a c ≃ 3.102, three saddle-node (SN) bifurcations occur simultaneously at the limit cycle, corresponding to an infinite-period transition in which C 3 symmetry is broken (since three stable attractors are created) [34]. As noted previously [32], this second transition is somewhat artificial from the perspective of more realistic models, for which one expects (and observes) oscillators to lock at a coupling-independent frequency.…”

Section: Mean Field Analysismentioning

confidence: 89%

“…As noted previously [32], this second transition is somewhat artificial from the perspective of more realistic models, for which one expects (and observes) oscillators to lock at a coupling-independent frequency. However, it is very interesting from the perspective of nonequilibrium phase transitions of interacting particle systems, since it provides a spontaneous breaking of C 3 symmetry in the absence of any absorbing state [34] (as opposed to models with 3 absorbing states, like rock-paper-scissors games [35][36][37][38][39]).…”

Section: Mean Field Analysismentioning

confidence: 99%

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Collective behavior of coupled nonuniform stochastic oscillators

Assis

Copelli

2012

Physica A: Statistical Mechanics and its Applications

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Theoretical studies of synchronization are usually based on models of coupled phase oscillators which, when isolated, have constant angular frequency. Stochastic discrete versions of these uniform oscillators have also appeared in the literature, with equal transition rates among the states. Here we start from the model recently introduced by Wood et al. [Phys. Rev. Lett. 96}, 145701 (2006)], which has a collectively synchronized phase, and parametrically modify the phase-coupled oscillators to render them (stochastically) nonuniform. We show that, depending on the nonuniformity parameter $0\leq \alpha \leq 1$, a mean field analysis predicts the occurrence of several phase transitions. In particular, the phase with collective oscillations is stable for the complete graph only for $\alpha \leq \alpha^\prime < 1$. At $\alpha=1$ the oscillators become excitable elements and the system has an absorbing state. In the excitable regime, no collective oscillations were found in the model.Comment: 17 pages, 4 figure

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Section: Mean Field Analysismentioning

confidence: 89%

Section: Mean Field Analysismentioning

confidence: 99%

Collective behavior of coupled nonuniform stochastic oscillators

Assis

Copelli

2012

Physica A: Statistical Mechanics and its Applications

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“…It is convenient to define an order parameter ψ that is identically zero (in the infinite-size limit) for a > a c . Assis et al proposed [15],…”

Section: Modelmentioning

confidence: 99%

“…(Color online) (From[15].) Order parameter |ψ| as a function of coupling a in mean-field theory and on the complete graph, for sizes as indicated.…”

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confidence: 99%

Synchronization of discrete oscillators on ring lattices and small-world networks

Rodrigues¹,

Dickman²

2020

J. Stat. Mech.

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A lattice of three-state stochastic phase-coupled oscillators introduced by Wood et al. [Phys. Rev. Lett. 96, 145701 (2006)] exhibits a phase transition at a critical value of the coupling parameter a, leading to stable global oscillations (GO). On a complete graph, upon further increase in a, the model exhibits an infinite-period (IP) phase transition, at which collective oscillations cease and discrete rotational (C 3 ) symmetry is broken [Assis et al., J. Stat. Mech. (2011) P09023]. These authors showed that the IP phase does not exist on finite-dimensional lattices. In the case of large negative values of the coupling, Escaff et al. [Phys. Rev. E 90, 052111(2014)] discovered the stability of travelling-wave states with no global synchronization but with local order. Here, we verify the IP phase in systems with long-range coupling but of lower connectivity than a complete graph and show that even for large positive coupling, the system sometimes fails to reach global order. The ensuing travelling-wave state appears to be a metastable configuration whose birth and decay (into the previously described phases) are associated with the initial conditions and fluctuations. PACS numbers: 64.60.Ht, 05.45.Xt, 89.75.-k arXiv:1912.04104v1 [cond-mat.stat-mech] 9 Dec 2019 2

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“…By "uniform global coupling" we mean all-to-all coupling all of equal strength. In a particular rendition of the model, further increasing the cou-pling strength leads to a slowing down of the state-tostate transitions until the oscillatory global state is lost altogether via an infinite-period bifurcation [24,25], and the system reaches a static stationary state in which most of the units are locked and static in the same state. Note that with uniform global coupling we do not introduce the notion of dimensionality, nor do we need to address any "spatial" questions.…”

Section: Introductionmentioning

confidence: 99%

Arrays of two-state stochastic oscillators: Roles of tail and range of interactions

et al. 2017

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We study the role of the tail and range of interaction in a spatially structured population of twostate on-off units governed by Markovian transition rates. The coupling among the oscillators is evidenced by the dependence of the transition rates of each unit on the states of the units to which it is coupled. Tuning the tail or range of the interactions, we observe a transition from an ordered global state (long range interactions) to a disordered one (short range interactions). Depending on the interaction kernel, the transition may be smooth (second order) or abrupt (first order). We analyze the transient, which may present different routes to the steady state with vastly different time scales.

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An infinite-period phase transition versus nucleation in a stochastic model of collective oscillations

Cited by 13 publications

References 27 publications

Collective behavior of coupled nonuniform stochastic oscillators

Collective behavior of coupled nonuniform stochastic oscillators

Synchronization of discrete oscillators on ring lattices and small-world networks

Arrays of two-state stochastic oscillators: Roles of tail and range of interactions

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