Stable chimeras of non-locally coupled Kuramoto–Sakaguchi oscillators in a finite array

Abstract

We consider chimera states of coupled identical phase oscillators where some oscillators are phase synchronized (have the same phase) while others are desynchronized. Chimera states of non-locally coupled Kuramoto–Sakaguchi oscillators in arrays of finite size are known to be chaotic transients; after a transient time, all the oscillators are phase synchronized, with the transient time increasing exponentially with the number of oscillators. In this work, we consider a small array of six non-locally coupled Kuramoto–Sakaguchi oscillators and modify the range of the phase lag parameter to destabilize their complete phase synchronization. Under these circumstances, we observe a chimera state spontaneously formed by the partition of oscillators into two independently synchronizable clusters of both stable and unstable synchronous states. In the chimera state, the trajectory of the phase differences of the desynchronized oscillators relative to the synchronous cluster is a stable periodic orbit, and as a result, the chimera state is a stable but not long-lived transient. We observe the chimera state with random initial conditions in a restricted range of the phase lag parameter and clarify why the state is observable in the restricted range using Floquet theory for periodic orbit stability.