Heterogeneity Induces Cyclops States in Kuramoto Networks with Higher-Mode Coupling

TL;DR

This paper demonstrates that heterogeneity in oscillator frequencies can stabilize complex multi-cluster states, like Cyclops states, in Kuramoto networks with higher-mode coupling, revealing a constructive role for disorder in network dynamics.

Contribution

It introduces a mesoscopic collective coordinate method linking microscopic heterogeneity to macroscopic stability, enabling the stabilization of Cyclops states through frequency diversity.

Findings

Heterogeneity stabilizes Cyclops and cluster states in Kuramoto networks.

Stable multi-state dynamics are achievable without external tuning.

A collective coordinate approach connects microscopic heterogeneity to stability.

Abstract

Disorder is often seen as detrimental to collective dynamics, yet recent work has shown that heterogeneity can enhance network synchronization. However, its constructive role in stabilizing nontrivial cooperative patterns remains largely unexplored. In this Letter, we show that frequency heterogeneity among oscillators can induce stable Cyclops and cluster states in Kuramoto networks with higher-mode coupling, even though these states are unstable in the identical oscillator case. Cyclops states, introduced in [Munyaev et al., Phys. Rev. Lett. 130, 107021 (2023)], feature two synchronized clusters and a solitary oscillator, requiring a delicate phase balance. Surprisingly, heterogeneity alone is sufficient to stabilize these patterns across a broad range of detuning values without needing to be compensated by other forms of disorder or external tuning. We introduce a mesoscopic collective coordinate approach that connects microscopic frequency structure, captured by the solitary oscillator's influence, with mean-field cluster-level stability. This constructive approach identifies favorable ranges of heterogeneity and suitable initial conditions for inducing robust multi-state dynamics, offering a foundation for their analysis in broader classes of heterogeneous biological and engineering networks.