Synchronous Rhythmic Flashing of Fireflies. II.

Buck, John D.

doi:10.1086/415929

Cited by 613 publications

(410 citation statements)

References 55 publications

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“…Example systems include pacemaker cells in the heart (Michaels et al, 1987), circadian cells in the brain (Liu et al, 1997), coupled cortical neurons (Crook et al, 1997), Hodgkin-Huxley neurons (Brown et al, 2003), brain networks (Varela et al, 2001), yeast cells (Ghosh et al, 1971), flashing fireflies (Buck, 1988;Ermentrout, 1991), chirping crickets (Walker, 1969), central pattern generators for animal locomotion (Kopell and Ermentrout, 1988), particle models mimicking animal flocking behavior (Ha et al, 2010b, and fish schools , among others. The coupled oscillator model (1) also appears in physics and chemistry in modeling and analysis of spin glass models (Daido, 1992;Jongen et al, 2001), flavor evolution of neutrinos (Pantaleone, 1998), coupled Josephson junctions (Wiesenfeld et al, 1998), coupled metronomes (Pantaleone, 2002), Huygen's coupled pendulum clocks (Bennett et al, 2002;Kapitaniak et al, 2012), micromechanical oscillators with optical (Zhang et al, 2012) or mechanical (Shim et al, 2007) coupling, and in the analysis of chemical oscillations (Kuramoto, 1984a;Kiss et al, 2002).…”

Section: Applications In Sciencesmentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.

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Section: Applications In Sciencesmentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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“…This many-body cooperative effect is observed in many physical and biological systems, pervading length and time scales of several orders of magnitude. Some examples are metabolic synchrony in yeast cell suspensions [1], synchronized firings of cardiac pacemaker cells [2], flashing in unison by groups of fireflies [3], voltage oscillations at a common frequency in an array of current-biased Josephson junctions [4], phase synchronization in electrical power distribution networks [5][6][7], rhythmic applause [8], animal flocking behavior [9]; see Ref. [10] for a recent survey.…”

Section: Introductionmentioning

confidence: 99%

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

2014

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We study the dynamics of a system of coupled oscillators of distributed natural frequencies, by including the features of both thermal noise, parametrized by a temperature, and inertial terms, parametrized by a moment of inertia. For a general unimodal frequency distribution, we report here the complete phase diagram of the model in the space of dimensionless moment of inertia, temperature, and width of the frequency distribution. We demonstrate that the system undergoes a nonequilibrium first-order phase transition from a synchronized phase at low parameter values to an incoherent phase at high values. We provide strong numerical evidence for the existence of both the synchronized and the incoherent phase, treating the latter analytically to obtain the corresponding linear stability threshold that bounds the first-order transition point from below. In the limit of zero noise and inertia, when the dynamics reduces to the one of the Kuramoto model, we recover the associated known continuous transition. At finite noise and inertia but in the absence of natural frequencies, the dynamics becomes that of a well-studied model of long-range interactions, the Hamiltonian mean-field model. Close to the first-order phase transition, we show that the escape time out of metastable states scales exponentially with the number of oscillators, which we explain to be stemming from the long-range nature of the interaction between the oscillators.

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“…Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics. [1,2,5,6,7,8], chemical [9,10], and physical systems [11,12]. However, even in this ideal limit, some basic questions including global, quantitative stability of asymptotic states, still remain open [13,14,15,16].…”

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confidence: 99%

Coarse Graining the Dynamics of Coupled Oscillators

2006

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We present an equation-free computational approach to the study of the coarse-grained dynamics of finite assemblies of non-identical coupled oscillators at and near full synchronization. We use coarse-grained observables which account for the (rapidly developing) correlations between phase angles and oscillator natural frequencies. Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics. [1,2,5,6,7,8], chemical [9,10], and physical systems [11,12]. However, even in this ideal limit, some basic questions including global, quantitative stability of asymptotic states, still remain open [13,14,15,16]. Many real-world systems consist of a large, finite number of non-identical entities, where statistical techniques for the continuum-limit are not directly applicable. Exploring and understanding the dynamics of such finite oscillator assemblies is an important topic (e.g., see Ref. [17]).We present a computer-assisted approach to modeling the coarse-grained dynamics of such large, finite oscillator assemblies at and near full synchronization. The premise is that there exist a small number of coarse-grained variables (observables) adequately describing the long-term dynamics, and that a closed evolution equation for these observables exists, but is not explicitly available. To account for oscillator variability within the assembly, we treat both the variable oscillator properties (here, natural frequencies ω) and the oscillator states (here, phase angles θ) as random variables. Recognizing a quick development of correlations between ω and θ, we express the latter as a polynomial expansion of the former (borrowing Wiener polynomial chaos (PC) tools [18]); the PC expansion coefficients are our coarse observables.Availability of the governing equations for the variables of interest is a prerequisite to modeling and computation. We circumvent this step using the recently-developed equation-free (EF) framework for complex, multiscale systems modeling [19,20,21]. In this framework we can perform system-level computational tasks without explicit knowledge of the coarse-grained equations; these unavailable equations are solved by designing, performing and processing the results of short bursts of appropriately initialized detailed (fine-scale, microscopic) simulations.We consider a paradigmatic model of coupled oscillators, the Kuramoto model, consisting of a population of N all-to-all, phase-coupled limit-cycle oscillators with i.i.d. ω with distribution function g(ω). This model has been extensively studied because of its simplicity and certain mathematical tractability, yet it is not merely a toy model. It appears as a normal form for general systems of coupled oscillators (e.g. Refs. [10,11]).We choose a Gaussian with standard deviation σ ω = 0.1 for g(ω); however, our approach is not limited to this particular choice, nor to the Kuramoto model. Due to rotational symmetry, the mean frequency Ω = i ω i /N can be...

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Synchronous Rhythmic Flashing of Fireflies. II.

Cited by 613 publications

References 55 publications

Synchronization in complex networks of phase oscillators: A survey

Synchronization in complex networks of phase oscillators: A survey

Nonequilibrium first-order phase transition in coupled oscillator systems with inertia and noise

Coarse Graining the Dynamics of Coupled Oscillators

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