Chimeras unfolded

Abstract

The instability of mixing in the Kuramoto model of coupled phase oscillators is the key to understanding a range of spatiotemporal patterns, which feature prominently in collective dynamics of systems ranging from neuronal networks, to coupled lasers, to power grids. In this paper, we describe a codimension-2 bifurcation of mixing whose unfolding, in addition to the classical scenario of the onset of synchronization, also explains the formation of clusters and chimeras. We use a combination of linear stability analysis and Penrose diagrams to identify and analyze a variety of spatiotemporal patterns including stationary and traveling coherent clusters and twisted states, as well as their combinations with regions of incoherent behavior called chimera states. The linear stability analysis is used to estimate of the velocity distribution within these structures. Penrose diagrams, on the other hand, predict accurately the basins of their existence. Furthermore, we show that network topology can endow chimera states with nontrivial spatial organization. In particular, we present twisted chimera states, whose coherent regions are organized as stationary or traveling twisted states. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model with uni-, bi-, and tri-modal frequency distributions and all-to-all and nonlocal nearest-neighbor connectivity.

arXiv comment18 pages, 7 figures

References in corpus (16)

• The Kuramoto model in complex networks
• Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
• Solvable model for chimera states of coupled oscillators
• Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators
• Exact Results for the Kuramoto Model with a Bimodal Frequency Distribution
• Coexistence of synchrony and incoherence in oscillatory media under nonlinear global coupling
• Chimera States in a Ring of Nonlocally Coupled Oscillators
• Amplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators
• The Spectrum of the Partially Locked State for the Kuramoto Model
• A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model
• Stability and bifurcation for the Kuramoto model
• The continuum limit of the Kuramoto model on sparse random graphs
• Bifurcations in the Kuramoto model on graphs
• Stability of clusters in the second-order Kuramoto model on random graphs
• Instability of mixing in the Kuramoto model: From bifurcations to patterns
• A Hopf bifurcation in the Kuramoto-Daido model

Cited by in corpus (3)

• Bifurcations and patterns in the Kuramoto model with inertia
• Stability of twisted states on lattices of Kuramoto oscillators
• Metastability in the stochastic nearest-neighbor Kuramoto model of coupled phase oscillators