On Computing the Critical Coupling Coefficient for the Kuramoto Model on a Complete Bipartite Graph

Verwoerd, Mark; Mason, Oliver

doi:10.1137/080725726

Cited by 69 publications

(63 citation statements)

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“…We remark that Verwoerd and Mason also extended the implicit formulae (44)- (45) to complete bipartite graphs (Verwoerd and Mason, 2009, Theorem 3) and infinite-dimensional networks (Verwoerd and Mason, 2011, Theorem 4). Moreover, they provided bisection algorithms to compute K critical with predefined precision in a finite number of iterations.…”

Section: Exact and Implicit Conditionsmentioning

confidence: 99%

“…In general, concise and accurate results are known only for specific topologies such as complete graphs (as discussed in the previous section), linear chains (Strogatz and Mirollo, 1988), highly symmetric ring graphs (Buzna et al, 2009), acyclic graphs (Dekker and Taylor, 2013), and complete bipartite graphs (Verwoerd and Mason, 2009) with uniform weights. For arbitrary coupling topologies, the literature contains only sufficient conditions Kumagai, 1980, 1982;Jadbabaie et al, 2004;Dörfler and Bullo, 2012b) as well as numerical and statistical investigations for large random net-11 More precisely, the incremental norms B T ω p are seminorms in R n and proper norms in the quotient space 1 ⊥ n .…”

Section: Survey Of Synchronization Metrics and Conditionsmentioning

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 Conditionsmentioning

confidence: 99%

Section: Survey Of Synchronization Metrics and Conditionsmentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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“…[18][19][20], necessary and sufficient conditions are given for fixed points to exist for the finite size Kuramoto model for complete and bipartite graphs, and explicit upper and lower bounds of K c (N ) for these systems were also computed, followed by providing an algorithm to compute K c (N ). For the complete graph, similar results were presented in [21,22], where it was additionally shown that there is exactly one single stable equilibrium for K > K c (N ).…”

mentioning

confidence: 99%

Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis

Mehta

Daleo

Dörfler

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system of equations, which is an important yet challenging problem. We translate this into an algebraic geometry problem and use numerical methods to find all of the equilibria for various choices of coupling constants K, natural frequencies, and on different graphs. We note that for even modest sizes (N ∼ 10–20), the number of equilibria is already more than 100 000. We analyze the stability of each computed equilibrium as well as the configuration of angles. Our exploration of the equilibrium landscape leads to unexpected and possibly surprising results including non-monotonicity in the number of equilibria, a predictable pattern in the indices of equilibria, counter-examples to conjectures, multi-stable equilibrium landscapes, scenarios with only unstable equilibria, and multiple distinct extrema in the stable equilibrium distribution as a function of the number of cycles in the graph.

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“…Verwoerd and Mason also extended their results to bipartite graphs [24] but did not carry out a stability analysis. The formulas (8)- (9) can be reduced exactly to the implicit selfconsistency equation derived by Mirollo and Strogatz in [20] and by Aeyels and Rogge in [21], where additionally a local stability analysis is carried out.…”

Section: Implicit and Exact Bounds For The Finite Dimensional Kuramentioning

confidence: 99%

“…. , n} and for all t ≥ 0 [9], [19], [21], [23], [24]. Other commonly used terms in the vast synchronization literature include full, exact, or perfect synchronization (or even phase locking [3]) for phase synchronization and frequency locking, frequency entrainment, or partial synchronization for frequency synchronization.…”

Section: The Kuramoto Model Of Coupled Oscillatorsmentioning

confidence: 99%

On the critical coupling strength for Kuramoto oscillators

Dörfler

Bullo

2011

Proceedings of the 2011 American Control Conference

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Abstract-The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made dynamical systems of coupled oscillators. It is well-known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features three contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength in the finite and infinitedimensional case. Finally, we present the first explicit necessary and sufficient condition on the critical coupling strength to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. I. THE KURAMOTO MODEL OF COUPLED OSCILLATORSA classic model for the synchronization of coupled oscillators is due to Kuramoto [1]. The Kuramoto model considers n ≥ 2 coupled oscillators each represented by a phase variable θ i ∈ T 1 , the 1-tours, and a natural frequency ω i ∈ R. The system of coupled oscillators obeys the dynamicṡwhere K > 0 is the coupling strength among the oscillators. The Kuramoto model (1) finds application in various biological synchronization phenomena, and we refer the reader to the excellent reviews Kuramoto himself analyzed the model (1) based on the order parameter re iψ 1 n n j=1 e iθj , which corresponds the centroid of all oscillators when represented as points on the unit circle in C 1 . The magnitude r of the order parameter can be understood as a measure of synchronization: if all oscillators are perfectly synchronized with identical angles θ i (t), then r = 1, and if all oscillators are spaced equally on the unit circle, then r = 0. With the order parameter, the Kuramoto model (1) can be written in the insightful forṁThis work was supported in part by NSF grants IIS-0904501 and CNS-0834446.Florian Dörfler and Francesco Bullo are with the Center for Control, Dynamical Systems and Computation, University of California at Santa Barbara, Santa Barbara, CA 93106, {dorfler, bullo}@engineering.ucsb.edu Equation (2) gives the intuition that the oscillators synchronize by coupling to a mean field represented by the order parameter re iψ . Intuitively, for small coupling strength K each oscillator rotates with its natural frequency ω i , whereas for large coupling strength K all angles θ i (t) will be entrained by the mean field re iψ . This phase transition from incoherency to synchronization occurs for some critical coupling K critical and has been the source of nume...

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On Computing the Critical Coupling Coefficient for the Kuramoto Model on a Complete Bipartite Graph

Cited by 69 publications

References 50 publications

Synchronization in complex networks of phase oscillators: A survey

Synchronization in complex networks of phase oscillators: A survey

Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis

On the critical coupling strength for Kuramoto oscillators

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