Complete synchronization of Kuramoto oscillators with finite inertia

Choi, Young Pil; Ha, Seung Yeal; Yun, Seok Bae

doi:10.1016/j.physd.2010.08.004

Cited by 119 publications

(97 citation statements)

References 24 publications

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“…The analysis of second-order oscillator networks has also received a lot of attention, see (Acebrón et al, 2005;Dörfler and Bullo, 2011;Choi et al, 2011b) for a literature overview. Among others, the contraction Lyapunov function (32) can be extended to second-order oscillators (Choi et al, 2011b), the continuum-limit analysis can be extended (Acebrón et al, 2005), and the local stability properties are preserved when going from first to second order (Dörfler and Bullo, 2011). Of course, the transient dynamics of second-order oscillator networks have their own characteristics, especially for large inertia and small damping (Paganini and Lesieutre, 1999).…”

Section: Conclusion and Open Research Directionsmentioning

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: Conclusion and Open Research Directionsmentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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“…One might devise many distinct dynamical systems for a given M and x 0 . To date, FRP has been studied for some special manifolds, such as the flat space R d via the Cucker-Smale flocking model, [9][10][11][12][13][14] the circle S 1 via the Kuramoto model, [15][16][17][18][19][20][21][22][23] and the infinite cylinder via a generalized Kuramoto model. 24 As far as the authors are aware, there is currently no general approach for FRP on an arbitrary Riemannian manifold M.…”

Section: Introductionmentioning

confidence: 99%

Emergent behaviors of a holonomic particle system on a sphere

Chi¹,

Choi²,

Ha³

2014

Journal of Mathematical Physics

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We study sufficient conditions for the asymptotic emergence of synchronous behaviors in a holonomic particle system on a sphere, which was recently introduced by Lohe [“Non-Abelian Kuramoto model and synchronization,” J. Phys. A: Math. Theor. 42, 395101–395126 (2009)]. These conditions depend only on the coupling strength and initial position diameter. For identical particles, we show that the position diameter approaches zero asymptotically under these sufficient conditions, i.e., all particles approach to the same position. For non-identical particles, the particle positions do not shrink to one point, but can be squeezed into some small region whose diameter is inversely proportional to the coupling strength, when the coupling strength is large. We also provide several numerical results to confirm our analytical findings.

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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%

“…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%

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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Complete synchronization of Kuramoto oscillators with finite inertia

Cited by 119 publications

References 24 publications

Synchronization in complex networks of phase oscillators: A survey

Synchronization in complex networks of phase oscillators: A survey

Emergent behaviors of a holonomic particle system on a sphere

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

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