Interplay of inertia and external force in Kuramoto model

Abstract

The impact of external force is well studied in the Kuramoto model without inertia but remains unclear for inertial Kuramoto model (KMI) with bimodal intrinsic frequency distributions. Filling this gap, we demonstrate that competition between external force and intrinsic bimodality can suppress the intermediate standing wave state of bimodal KMI by entraining oscillators to the external force. Using a self-consistent analytical framework, we show that forcing makes the backward transition discontinuous, unlike the continuous transition in the unimodal case. Furthermore, for a bi-delta distribution, we derive a closed-form expression for the backward solution branch. These results demonstrate how intrinsic frequency structure shapes the effect of external force with possible implications in biological systems (e.g., photoreceptor and pacemaker cells) and for pinning-control strategies in multi-agent networks. © 2026 Author(s).