Gluing Kuramoto coupled oscillators networks

Canale, Eduardo; Monzón, Pablo

doi:10.1109/cdc.2007.4434382

Cited by 15 publications

(15 citation statements)

References 20 publications

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“…While previous research has emphasized the distinction between linear and nonlinear spaces [Sarlette and Sepulchre, 2009;Tron et al, 2013], it has not resulted in further categorization. Still, much has been revealed with respect to specific manifolds, in particular for special instances of the Stiefel manifold, see e.g., [Canale and Monzón, 2007;Markdahl and Gonc ¸alves, 2016;DeVille, 2018]. This paper unifies such results by connecting them to the existence or non-existence of a closed curve of locally minimum length in the manifold.…”

Section: Introductionmentioning

confidence: 73%

A Geometric Obstruction to Almost Global Synchronization on Riemannian Manifolds

Markdahl

2018

Preprint

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Multi-agent systems on nonlinear spaces sometimes fail to synchronize. This is usually attributed to the initial configuration of the agents being too spread out, the graph topology having certain undesired symmetries, or both. Besides nonlinearity, the role played by the geometry and topology of the nonlinear space is often overlooked. This paper concerns two gradient descent flows of quadratic disagreement functions on general Riemannian manifolds. One system is intrinsic while the other is extrinsic. We derive necessary conditions for the agents to synchronize from almost all initial conditions when the graph used to model the network is connected. If a Riemannian manifold contains a closed curve of locally minimum length, then there is a connected graph and a dense set of initial conditions from which the intrinsic system fails to synchronize. The extrinsic system fails to synchronize if the manifold is multiply connected. The extrinsic system appears in the Kuramoto model on S 1 , rigid-body attitude synchronization on SO(3), the Lohe model of quantum synchronization on the n-sphere, and the Lohe model on U(n). Except for the Lohe model on the n-sphere where n ∈ N\{1}, there are dense sets of initial conditions on which these systems fail to synchronize. The reason for this difference is that the n-sphere is simply connected for all n ∈ N\{1} whereas the other manifolds are multiply connected.

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Section: Introductionmentioning

confidence: 73%

A Geometric Obstruction to Almost Global Synchronization on Riemannian Manifolds

Markdahl

2018

Preprint

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“…then the trajectories will be also bounded since the phase average remains the same (for system (2):φ 1 + · · · +φ n = 0). The PIS, therefore, is defined through the maximum phase difference that is bounded by the value of D: (5) which is obviously a compact.…”

Section: A Preliminary Resultsmentioning

confidence: 99%

Synchronization of heterogeneous Kuramoto oscillators with graphs of diameter two

Gushchin

Mallada

Tang

2015

2015 49th Annual Conference on Information Sciences and Systems (CISS)

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Abstract-In this article we study synchronization of Kuramoto oscillators with heterogeneous frequencies, and where underlying topology is a graph of diameter two. When the coupling strengths between every two connected oscillators are the same, we find an analytic condition that guarantees an existence of a Positively Invariant Set (PIS) and demonstrate that existence of a PIS suffices for frequency synchronization. For graphs of diameter two, this synchronization condition is significantly better than existing general conditions for an arbitrary topology. If the coupling strengths can be different for different pairs of connected oscillators, we formulate an optimization problem that finds sufficient for synchronization coupling strengths such that their sum is minimal.

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“…Recently, in [5], [6], we made some progress proving that: 1) A graph synchronizes if and only if its blocks do as well.…”

Section: Discussionmentioning

confidence: 99%

The wheels: an infinite family of bi-connected planar synchronizing graphs

Canale

Monzón

Robledo

2010

2010 5th IEEE Conference on Industrial Electronics and Applications

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Almost global synchronization property of Kuramoto coupled oscillations was recently introduced and stands for the case where almost every initial condition of the dynamical system leads to the synchronization of all the agents. When the oscillators are all identical, the property only depends on the the underlying interconnection graph. If the property is present, the interconnection graph is called synchronizing. It is known that a graph is synchronizing if and only if its block are. So, the characterization of synchronizing graphs can be restricted to the class of bi-connected graphs. In this work, we present the first known infinite family of biconnected planar synchronizing graphs, named, the wheels. Besides, we present two graph wich are the first known chordal graph and Halin graphs that not synchronize.

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Gluing Kuramoto coupled oscillators networks

Abstract: Abstract-In this work we prove that the problem of almost global synchronization of the Kuramoto model of sinusoidally symmetric coupled oscillators with a given topology could be reduced to the analysis of the blocks of the underlying interconnection graph.

Cited by 15 publications

References 20 publications

A Geometric Obstruction to Almost Global Synchronization on Riemannian Manifolds

A Geometric Obstruction to Almost Global Synchronization on Riemannian Manifolds

Synchronization of heterogeneous Kuramoto oscillators with graphs of diameter two

The wheels: an infinite family of bi-connected planar synchronizing graphs

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