Almost Global Synchronization of Symmetric Kuramoto Coupled Oscillators

Canale, Eduardo; Monzón, Pablo

doi:10.5772/6026

Cited by 24 publications

(27 citation statements)

References 21 publications

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“…Of particular interest are so-called S 1 -synchronizing graphs for which all critical points of (25) are hyperbolic, the phasesynchronized state is the global minimum of U (θ), and all other critical points are local maxima or saddle points. The class of S 1 -synchronizing graphs includes, among others, complete graphs and acyclic graphs (Monzón and Paganini, 2005;Canale and Monzón, 2008;Sarlette, 2009;Canale et al, 2010b,a). These basic results motivated the study of the critical points and of the curvature of the potential energy U (θ) in the literature on the theory and applications of synchronization, including, the study of transient stability in power systems and the design of motion coordination controllers for planar vehicles, see Subsections 2.1 and 2.2.…”

Section: Potential Landscape Analysismentioning

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: Potential Landscape Analysismentioning

confidence: 99%

Synchronization in complex networks of phase oscillators: A survey

2014

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“…. , 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%

“…In this case, the analysis of the Kuramoto model (1) is particularly simple and almost global stability can be derived by various methods. A sample of different analysis schemes includes the contraction property [22], quadratic Lyapunov functions [13], linearization [23], or order parameter and potential function arguments [4].…”

Section: Review Of Bounds For the Critical Couplingmentioning

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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“…Over the years, different types of synchronization have been characterized, such as the complete synchronization, lag synchronization, generalized synchronization, phase and imperfect phase synchronization [Srinivasan et al, 2012;Wu et al, 2012;Suresh et al, 2012]. In this paper, we also consider the almost global synchronization [Canale & Monzón, 2008].…”

Section: Introductionmentioning

confidence: 99%

“…First, let us recall that complete synchronization can be defined as the equality of the state variables while evolving in time, in the case of coupled identical systems. We also consider the notion of almost global synchronization [Canale & Monzón, 2008], which is defined as the case in which almost all initial conditions (except at most a set of measure zero) in the state space give rise to iterated sequences converging towards the set where the variables of the state space are equal, the diagonal set. For this aim, we consider Lyapunov functions and we study more precisely the eigenvalues and associated eigenvectors on a direction transverse to the diagonal.…”

Section: Introductionmentioning

confidence: 99%

Synchronization and Basins of Synchronized States in Two-Dimensional Piecewise Maps via Coupling Three Pieces of One-Dimensional Maps

Fournier-Prunaret¹,

Rocha

Caneco

et al. 2013

Int. J. Bifurcation Chaos

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This paper is devoted to the synchronization of a dynamical system defined by two different coupling versions of two identical piecewise linear bimodal maps. We consider both local and global studies, using different tools as natural transversal Lyapunov exponent, Lyapunov functions, eigenvalues and eigenvectors and numerical simulations. We obtain theoretical results for the existence of synchronization on coupling parameter range. We characterize the synchronization manifold as an attractor and measure the synchronization speed. In one coupling version, we give a necessary and sufficient condition for the synchronization. We study the basins of synchronization and show that, depending upon the type of coupling, they can have very different shapes and are not necessarily constituted by the whole phase space; in some cases, they can be riddled.

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Almost Global Synchronization of Symmetric Kuramoto Coupled Oscillators

Cited by 24 publications

References 21 publications

Synchronization in complex networks of phase oscillators: A survey

Synchronization in complex networks of phase oscillators: A survey

On the critical coupling strength for Kuramoto oscillators

Synchronization and Basins of Synchronized States in Two-Dimensional Piecewise Maps via Coupling Three Pieces of One-Dimensional Maps

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