Bistability of patterns of synchrony in Kuramoto oscillators with inertia

We study the co-existence of stable patterns of synchrony in two coupled populations of identical Kuramoto oscillators with inertia. The two populations have different sizes and can split into two clusters where the oscillators synchronize within a cluster while there is a phase shift between the dynamics of the two clusters. Due to the presence of inertia, which increases the dimensionality of the oscillator dynamics, this phase shift can oscillate, inducing a breathing cluster pattern. We derive analytical conditions for the co-existence of stable two-cluster patterns with constant and oscillating phase shifts. We demonstrate that the dynamics, that governs the bistability of the phase shifts, is described by a driven pendulum equation. We also discuss the implications of our stability results to the stability of chimeras.