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

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• 作者：Georgi S. Medvedev ; Xuezhi Tang
• 关键词：Kuramoto model ; Twisted state ; Synchronization ; Quasirandom graph ; Cayley graph ; Paley graph ; 34C15 ; 45J05 ; 45L05 ; 05C90
• 刊名：Journal of Nonlinear Science
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：25
• 期：6
• 页码：1169-1208
• 全文大小：1,317 KB
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• 作者单位：Georgi S. Medvedev (1)
  Xuezhi Tang (1)
  
  1. Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA, 19104, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Mathematical and Computational Physics
  Mechanics
  Applied Mathematics and Computational Methods of Engineering
  Economic Theory
  
• 出版者：Springer New York
• ISSN：1432-1467

文摘

The Kuramoto model of coupled phase oscillators on complete, Paley, and Erd?s–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. Specifically, we identify twisted states, steady-state solutions of the Kuramoto model on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the initial value problems for the Kuramoto model on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the Kuramoto model on Cayley and random graphs. Keywords Kuramoto model Twisted state Synchronization Quasirandom graph Cayley graph Paley graph