Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model

Choi, Young Pil; Ha, Seung Yeal; Jung, Sung‐Kyun; Kim, Yongduck

doi:10.1016/j.physd.2011.11.011

Cited by 195 publications

(157 citation statements)

References 29 publications

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“…We remark that similar analysis results are reported in (De Smet and Aeyels, 2007;Ha et al, 2010a;Choi et al, 2011a;Dörfler and Bullo, 2012b;Schmidt et al, 2012), and the bound γ min on the ultimate phase distances can be improved for particular pairs of oscillators, see (Choi et al, 2011a, Theorem 5.2). Finally, we remark that the proof strategy via the contraction Lyapunov function (32) can be adapted to more general cases, for example, the conclusions of Theorem 6.6 can be extended to time-varying natural frequencies, see (Dörfler and Bullo, 2011) and the illustration in Fig. 14.…”

Section: Exact and Implicit Conditionssupporting

confidence: 80%

“…In the finite-dimensional case, various necessary, sufficient, implicit, and explicit estimates of the critical coupling strength K critical have been proposed (Hemmen and Wreszinski, 1993;Aeyels and Rogge, 2004;Jadbabaie et al, 2004;Acebrón et al, 2005;Mirollo and Strogatz, 2005;De Smet and Aeyels, 2007;Chopra and Spong, 2009;Mason, 2008, 2009;Chung and Slotine, 2010;Ha et al, 2010a;Choi et al, 2011a;Franci et al, 2011;Bullo, 2011, 2012b;Schmidt et al, 2012). We refer to (Dörfler and Bullo, 2011) for a comprehensive historical overview and present only the best known results here.…”

Section: Finite-dimensional Kuramoto Modelsmentioning

confidence: 99%

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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: Exact and Implicit Conditionssupporting

confidence: 80%

Section: Finite-dimensional Kuramoto Modelsmentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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“…For the supercritical case K > D(Ω), the synchronization estimates of the system (1.1) were studied in [4,5]. For the mean-field case (N → ∞), the linearized stability of the phase-locked state was investigated in [17,18,23,24].…”

Section: Discussion Of the Main Resultsmentioning

confidence: 99%

“…As observed in [5,8,11], for a finitedimensional Kuramoto model (1.1) with N < ∞, a new threshold coupling strength K = D(Ω) appears, when the complete frequency synchronization (in short CFS) of (1.1) with N < ∞ is considered (see Definition 2.1). In fact, Dörfler and Bullo [8] showed that if K ≤ D(Ω), then there exist a set of natural frequencies {Ω 1 ,···Ω N } and initial configuration θ 0 ,…”

Section: Introductionmentioning

confidence: 84%

“…Thus, throughout the paper, we use the terminologies "super-critical " and "critical " regimes to denote the regimes K > D(Ω) and K = D(Ω), respectively. In [4,5,6,8,9,11,13], the CFS was extensively studied in the super-critical regime K > D(Ω). In contrast, the corresponding synchronization problem has not been well studied in the critical regime where K = D(Ω).…”

Section: Introductionmentioning

confidence: 99%

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Exponential synchronization of finite-dimensional Kuramoto model at critical coupling strength

Choi

Kang

et al. 2013

Communications in Mathematical Sciences

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Abstract. We discuss the exponential synchronization for an ensemble of Kuramoto oscillators at the critical coupling strength, which is the diameter of the set consisting of natural frequencies. When the number of distinct natural frequencies is greater than two and the initial phases are strictly confined in an interval of length π 2 , we show that the initial configuration evolves toward a phase-locked state at least exponentially fast. This fast convergence toward the phase-locked state is markedly different from an ensemble of Kuramoto oscillators with only two distinct natural frequencies. For this, we derive a Gronwall inequality for the frequency diameter to obtain complete synchronization. We also compare our analytical results with numerical simulation results.

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Landau Damping to Partially Locked States in the Kuramoto Model

Dietert

Fernandez

Gérard-Varet

2018

Comm Pure Appl Math

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In the Kuramoto model of globally coupled oscillators, partially locked states (PLS) are stationary solutions that capture the emergence of partial synchrony when the interaction strength increases. While PLS have long been considered, existing results on their stability are limited to neutral stability of the linearized dynamics in strong topology or to specific invariant subspaces (obtained via the so‐called Ott‐Antonsen (OA) ansatz) with specific frequency distributions for the oscillators. In the mean‐field limit, the Kuramoto model shows various ingredients of the Landau damping mechanism in the Vlasov equation. This analogy has been a source of inspiration for stability proofs of regular Kuramoto equilibria. In addition, the major mathematical issue with PLS asymptotic stability is that these states consist of heterogeneous and singular measures. Here we establish an explicit criterion for their spectral stability and prove their local asymptotic stability in weak topology for a large class of analytic frequency marginals. The proof strongly relies on a suitable functional space that contains (Fourier transforms of) singular measures, and for which the linearized dynamics is well under control. For illustration, the stability criterion is evaluated in some standard examples. We confirm in particular that no loss of generality results in assuming the OA ansatz. To the best of our knowledge, our result provides the first proof of Landau damping to heterogeneous and irregular equilibria in the absence of dissipation. © 2018 Wiley Periodicals, Inc.

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Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model

Cited by 195 publications

References 29 publications

Synchronization in complex networks of phase oscillators: A survey

Synchronization in complex networks of phase oscillators: A survey

Exponential synchronization of finite-dimensional Kuramoto model at critical coupling strength

Landau Damping to Partially Locked States in the Kuramoto Model

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