Dynamics of fully coupled rotators with unimodal and bimodal frequency distribution

TL;DR

This paper investigates the synchronization transitions in a network of coupled rotators with inertia, revealing complex hysteretic behaviors and state coexistence in unimodal and bimodal frequency distributions, with implications for large systems.

Contribution

It provides a detailed analysis of how inertia and frequency distribution shape the hysteretic synchronization transitions and state coexistence in coupled rotator networks.

Findings

Discontinuous hysteretic transition in unimodal case.

Coexistence of partially synchronized and standing wave states for large inertia.

Multiple states including traveling waves and partially synchronized regimes in bimodal case.

Abstract

We analyze the synchronization transition of a globally coupled network of N phase oscillators with inertia (rotators) whose natural frequencies are unimodally or bimodally distributed. In the unimodal case, the system exhibits a discontinuous hysteretic transition from an incoherent to a partially synchronized (PS) state. For sufficiently large inertia, the system reveals the coexistence of a PS state and of a standing wave (SW) solution. In the bimodal case, the hysteretic synchronization transition involves several states. Namely, the system becomes coherent passing through traveling waves (TWs), SWs and finally arriving to a PS regime. The transition to the PS state from the SW occurs always at the same coupling, independently of the system size, while its value increases linearly with the inertia. On the other hand the critical coupling required to observe TWs and SWs increases with N suggesting that in the thermodynamic limit the transition from incoherence to PS will occur without any intermediate states. Finally a linear stability analysis reveals that the system is hysteretic not only at the level of macroscopic indicators, but also microscopically as verified by measuring the maximal Lyapunov exponent.