Transverse Lyapunov Exponent and Chimeras in Globally Coupled Maps

Abstract

We study the stability properties and the long-term dynamics of chimeras in systems of globally coupled maps. In particular, we establish a formula for the transverse Lyapunov exponent of the states of the system containing synchronized units. We use this formula to present numerical evidence of attracting chimeras having chaotic dynamics, as well as periodic behaviors. We also show that, at least for polynomial local maps, attracting periodic cycles tend to belong to cluster spaces and that, more generally, the limit sets of chimeras orbits have zero Lebesgue measures for strong coupling regimes.