Bifurcations and singularities for coupled oscillators with inertia and frustration

TL;DR

This paper demonstrates that even minimal inertia can fundamentally alter the synchronization transition in coupled oscillator models, with singularities near bifurcations playing a crucial role in system behavior.

Contribution

It reveals how small inertia influences the nature of synchronization transitions in Kuramoto-like models through an unstable manifold expansion approach.

Findings

Inertia changes the transition from continuous to discontinuous or vice versa.

Singularities near bifurcations control the qualitative dynamics.

Numerical tests support the theoretical predictions.

Abstract

We prove that any non zero inertia, however small, is able to change the nature of the synchronization transition in Kuramoto-like models, either from continuous to discontinuous, or from discontinuous to continuous. This result is obtained through an unstable manifold expansion in the spirit of J.D. Crawford, which features singularities in the vicinity of the bifurcation. Far from being unwanted artifacts, these singularities actually control the qualitative behavior of the system. Our numerical tests fully support this picture.