Stability of twisted states in the Kuramoto model on Cayley and random graphs
arXiv:1407.3990 · doi:10.1007/s00332-015-9252-y
Abstract
The Kuramoto model (KM) of coupled phase oscillators on complete, Paley, and Erdos-Renyi (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 KM on these graphs can be qualitatively different. Specifically, we identify twisted states, steady state solutions of the KM 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 IVPs for the KM 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 KM on Cayley and random graphs.
arXiv commentJournal of Nonlinear Science, 2015
References in corpus (5)
- Chimera States in a Ring of Nonlocally Coupled Oscillators
- The nonlinear heat equation on W-random graphs
- Fully Synchronous Solutions and the Synchronization Phase Transition for the Finite N Kuramoto Model
- Small-world networks of Kuramoto oscillators
- The geometry of spontaneous spiking in neuronal networks
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- Phase Oscillator Networks with Nonlocal Higher-Order Interactions: Twisted States, Stability and Bifurcations
- When is sync globally stable in sparse networks of identical Kuramoto oscillators?
- Dynamical Systems on Graph Limits and Their Symmetries
- Hopf Bifurcations of Twisted States in Phase Oscillators Rings with Nonpairwise Higher-Order Interactions
- Stability of twisted states on lattices of Kuramoto oscillators
- Metastability in the stochastic nearest-neighbor Kuramoto model of coupled phase oscillators