Chimerapedia: coherence–incoherence patterns in one, two and three dimensions

Abstract

Chimera states, or coherence–incoherence patterns in systems of symmetrically coupled identical oscillators, have been the subject of intensive study for the last two decades. In particular it is now known that the continuum limit of phase-coupled oscillators allows an elegant mathematical description of these states based on a nonlinear integro-differential equation known as the Ott–Antonsen equation. However, a systematic study of this equation usually requires a substantial computational effort. In this paper, we consider a special class of nonlocally coupled phase oscillator models where the above analytical approach simplifies significantly, leading to a semi-analytical description of both chimera states and of their linear stability properties. We apply this approach to phase oscillators on a one-dimensional lattice, on a two-dimensional square lattice and on a three-dimensional cubic lattice, all three with periodic boundary conditions. For each of these systems we identify multiple symmetric coherence–incoherence patterns and compute their linear stability properties. In addition, we describe how chimera states in higher-dimensional models are inherited from lower-dimensional models and explain how they can be grouped according to their symmetry properties and global order parameter.