Breathing and Rotobreathing Cyclops States in Phase Oscillators with Inertia and Two-Harmonic Coupling

Abstract

Cyclops states - three-cluster configurations consisting of two synchronous groups and a solitary oscillator - dominate in ensembles of phase oscillators with inertia and multiple coupling harmonics [Phys. Rev. E 109, 054202 (2024)]. In this work, for the first time, we systematically study nonstationary cyclops states that preserve the three-cluster structure: breathing and rotobreathing cyclops states. We identify two scenarios for their destabilization: period doubling, leading to quasicyclops states while preserving frequency synchronization within the clusters, and the destruction of one or two clusters, resulting in the emergence of switching cyclops or multicluster states. We show that breathing and rotobreathing cyclops states occupy vast parameter regions of the second coupling harmonic and are key elements of the dynamics. The results are important for predicting and controlling complex collective states in ensembles with higher-order interaction harmonics of various natures.