Oscillatory states, standing waves, and additional synchronized clusters in networks of second-order oscillators: uncovering the role of inertia

TL;DR

This paper explores how inertia in second-order oscillator networks leads to complex synchronized states, including oscillatory, standing wave, and secondary clusters, through a bistable mechanism and mean-field analysis.

Contribution

It introduces a novel explanation for secondary synchronized clusters induced by inertia, using a time-periodic mean-field approach and simplified models.

Findings

Inertia causes secondary synchronized clusters during synchronization transition.

Bistability between fixed points and invariant curves explains cluster formation.

Inertias weaken the influence of giant synchronized clusters on individual oscillators.

Abstract

We discuss the appearance of oscillatory and standing wave states in second-order oscillator networks showing that it is a special case of a more general mechanism involving secondary synchronized clusters induced by inertia. Using a time-periodic mean-field ansatz, we find a bistable mechanism involving a stable fixed point and an invariant curve of an appropriate Poincar\'e map. The bistability and the devil's staircase associated to the rotation number on the invariant curve provide an explanation for the appearance of the secondary synchronized clusters. The effect of inertias in the self-organization process is analyzed through a simplified model. This shows that the effect of giant synchronized clusters on the other oscillators is weakened by inertias, thus leading to secondary synchronized clusters during the transition process to synchronization.