Stability of rotatory solitary states in Kuramoto networks with inertia

TL;DR

This paper analyzes the stability of solitary states in Kuramoto oscillator networks with inertia, revealing conditions under which these states are stable, influenced by network size, inertia, and coupling type, with implications for rotatory chimeras.

Contribution

It provides the first asymptotic stability conditions for solitary states in inertial Kuramoto networks with phase lag, highlighting counterintuitive stability phenomena.

Findings

Increasing cluster size can stabilize solitary states in attractive coupling.

Solitary states can be stable in small networks with repulsive coupling.

Inertia induces rotatory dynamics affecting solitary state stability.

Abstract

Solitary states emerge in oscillator networks when one oscillator separates from the fully synchronized cluster and becomes incoherent with the rest of the network. Such chimera-type patterns with an incoherent state formed by a single oscillator were observed in various oscillator networks; however, there is still a lack of understanding of how such states can stably appear. Here, we study the stability of solitary states in Kuramoto networks of identical two-dimensional phase oscillators with inertia and a phase-lagged coupling. The presence of inertia can induce rotatory dynamics of the phase difference between the solitary oscillator and the coherent cluster. We derive asymptotic stability conditions for such a solitary state as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (i) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (ii) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our stability analysis for the emergence of rotatory chimeras.