Interplay of inertia and external forcing in Kuramoto model

TL;DR

This paper investigates how external forcing influences inertial Kuramoto oscillators with bimodal frequency distributions, revealing that forcing can suppress certain states and alter transition types.

Contribution

It provides a self-consistent analytical framework for understanding external forcing effects on bimodal inertial Kuramoto oscillators, including a closed-form solution for specific distributions.

Findings

Forcing suppresses intermediate standing wave states in bimodal KMI.

Forcing causes the backward transition to become discontinuous in bimodal distributions.

Derived a closed-form expression for the backward solution branch in bi-delta distributions.

Abstract

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