Journal
JOURNAL OF NONLINEAR SCIENCE
Volume 25, Issue 6, Pages 1169-1208Publisher
SPRINGER
DOI: 10.1007/s00332-015-9252-y
Keywords
Kuramoto model; Twisted state; Synchronization; Quasirandom graph; Cayley graph; Paley graph
Categories
Funding
- NSF [DMS 1109367, DMS 1412066]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1412066] Funding Source: National Science Foundation
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The Kuramoto model of coupled phase oscillators on complete, Paley, and ErdAs-R,nyi (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.
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