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The Kuramoto Model on Power Law Graphs: Synchronization and Contrast States

Abstract

The relation between the structural properties of the network and its dynamics is a central question in the analysis of dynamical networks. It is especially relevant for complex networks found in real-world applications. This work presents mathematically rigorous analysis of coupled dynamical systems on power law graphs. Specifically, we study large systems of coupled Kuramoto phase oscillators. In the limit as the size of the network tends to infinity, we derive analytically tractable mean field partial differential equation for the probability density function describing the state of the coupled system. The mean field limit is used to establish an explicit formula for the synchronization threshold for coupled phase oscillators with randomly distributed intrinsic frequencies. Furthermore, we study stable spatial patterns generated by the Kuramoto model with repulsive coupling. In particular, we identify a family of stable steady-state solutions having multiple regions with distinct statistical properties. We call these solutions contrast states. Like chimera states, contrast states exhibit coexisting regions of highly localized (coherent) behavior and highly irregular (incoherent) distribution of phases. We provide a detailed mathematical analysis of contrast states in the KM using the Ott–Antonsen ansatz. The analysis of synchronization and contrast states provides new insights into the role of power law connectivity in shaping dynamics of coupled dynamical systems. In particular, we show 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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Notes

  1. Throughout this paper, we use \(a\wedge b\) and \(a\vee b\) to denote \(\min \{a,b\}\) and \(\max \{a,b\}\), respectively.

  2. Here and below, \({\mathbb E~}_\omega \) denotes the mathematical expectation with respect to the probability space (2.4) underlying the random graph model.

  3. Equation (2.6) is derived from a system of weakly coupled oscillators (Kuramoto 1975; Hoppensteadt and Izhikevich 1997).

  4. See Guckenheimer and Holmes (1990) for the definition of the \(\omega \)–limit set.

  5. This order parameter is used in the analysis of the bifurcation underlying the transition to synchrony in the Kuramoto model on graphs (Chiba and Medvedev 2016, 2017). A similar order parameter was used by Laing for the analysis of chimera states in a ring of coupled oscillators (Laing 2009).

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Acknowledgements

This work was supported in part by the NSF DMS grants 1412066 and 1715161 (to GM).

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Correspondence to Georgi S. Medvedev.

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Communicated by Paul Newton.

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Medvedev, G.S., Tang, X. The Kuramoto Model on Power Law Graphs: Synchronization and Contrast States. J Nonlinear Sci 30, 2405–2427 (2020). https://doi.org/10.1007/s00332-018-9489-3

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Keywords

  • Coupled oscillators
  • Synchronization
  • Chimera state
  • Scale free graph
  • Graph limit
  • Mean field limit