Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

  title={Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs},
  author={Georgi S. Medvedev and Xuezhi Tang},
  journal={Journal of Nonlinear Science},
The Kuramoto model of coupled phase oscillators on complete, Paley, and Erdős–Rényi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the Kuramoto model on these graphs can be qualitatively different… 
The continuum limit of the Kuramoto model on sparse random graphs
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The averaging principle for approximating a dynamical system on a random graph by its deterministic (averaged) counterpart is proved and yields almost sure convergence on time intervals of order $\log n,$ where $n$ is the number of vertices.
Bifurcations in the Kuramoto model on graphs.
This work studies several model problems illustrating the link between network topology and synchronization in coupled dynamical systems, and identifies several families of graphs for which the transition to synchronization in the Kuramoto model starts at the same critical value of the coupling strength and proceeds in a similar manner.
The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations
In our previous work [Chiba, Medvedev, arXiv:1612.06493], we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on
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It is shown that despite sparse connectivity, power law networks possess remarkable synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law distribution.
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This work studies lattices of coupled Kuramoto oscillators with non-local interactions with a focus on the stability of twisted states, and exploits the "almost circulant" nature of the Jacobian to obtain a surprisingly accurate numerical test for stability.
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Consider n identical Kuramoto oscillators on a random graph. Specifically, consider Erdős–Rényi random graphs in which any two oscillators are bidirectionally coupled with unit strength,
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. The Kuramoto model provides a prototypical framework to synchronization phenomena in interacting particle systems. Apart from full phase synchrony where all oscillators behave identically,


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Multistability of twisted states in non-locally coupled Kuramoto-type models.
It is shown that the number of different stable multi-twisted states grows exponentially as N → ∞, and it is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable plays the role of time.
Stochastic Stability of Continuous Time Consensus Protocols
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It is pointed out that expanders, sparse highly connected graphs, generate CPs whose performance remains uniformly high when the size of the network grows unboundedly, and the benefits of using random versus regular network topologies for CP design are highlighted.
Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model.
A detailed analysis of the stability of phase-locked solutions to the Kuramoto system of oscillators is presented, including analytic expressions for the first and last frequency vectors to phase-locks, upper and lower bounds on the probability that a randomly chosen frequency vector will phase-lock, and very sharp results on the large N limit of this model.
The Nonlinear Heat Equation on W-Random Graphs
For systems of coupled differential equations on a sequence of W-random graphs, we derive the continuum limit in the form of an evolution integral equation. We prove that solutions of the initial
Spectral graph theory
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With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall
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A Chernoff inequality for matrices is used to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent upon the minimum and maximum expected degrees.
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The mathematical theory of pattern formation in electrically coupled networks of excitable neurons forced by small noise is presented and it is shown that the location of the minima of a certain continuous function on the surface of the unit n-cube encodes the most likely activity patterns generated by the network.
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The book Large Networks and Graph Limits, xiv + 475 pp., published in late 2012, comprises five parts, the first an illuminating introduction and the last a tantalizing taste of how the scope of the