Chimeras unfolded

TL;DR

This paper investigates the bifurcation mechanisms behind the emergence of complex spatiotemporal patterns, including chimeras, in the Kuramoto model, using stability analysis and Penrose diagrams to predict and analyze these phenomena.

Contribution

It introduces a codimension-2 bifurcation framework that explains the formation of chimeras and clusters, extending classical synchronization theory with new analytical tools.

Findings

Identification of various stable and traveling coherent structures.

Prediction of basins of existence for different patterns.

Demonstration of network topology effects on chimera organization.

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.