Self-consistent Method and Steady States of Second-order Oscillators

TL;DR

This paper improves the self-consistent method for analyzing steady states of second-order oscillators, especially in the small inertia regime, providing more accurate results near incoherence and introducing new concepts like margin region and scaled inertia.

Contribution

It presents a simplified and more accurate self-consistent analysis for second-order oscillators, addressing bistability and boundary complexities.

Findings

Enhanced accuracy near incoherence

Introduction of margin region and scaled inertia

More precise estimate of critical coupling

Abstract

The self-consistent method, first introduced by Kuramoto, is a powerful tool for the analysis of the steady states of coupled oscillator networks. For second-order oscillator networks complications to the application of the self-consistent method arise because of the bistable behavior due to the co-existence of a stable fixed point and a stable limit cycle, and the resulting complicated boundary between the corresponding basins of attraction. In this paper, we report on a self-consistent analysis of second-order oscillators which is simpler compared to previous approaches while giving more accurate results in the small inertia regime and close to incoherence. We apply the method to analyze the steady states of coupled second-order oscillators and we introduce the concepts of margin region and scaled inertia. The improved accuracy of the self-consistent method close to incoherence leads to an accurate estimate of the critical coupling corresponding to transitions from incoherence.