Stability Boundaries in Second-order Time-delayed Networks with Symmetry

TL;DR

This paper investigates the stability boundaries of second-order oscillator networks with symmetry, extending known criteria to second-order cases and analyzing a phase-locked loop network as an example.

Contribution

It extends the simple stability criterion from first-order to second-order oscillators with symmetry and analyzes stability boundaries in a specific network example.

Findings

Stability depends on the sign of the coupling function derivative for second-order oscillators.

Derived a simple stability criterion applicable to generic coupling functions.

Analyzed stability boundaries for a multi-node phase-locked loop network.

Abstract

In this contribution we aim to study the stability boundaries of solutions at equilibria for a second-order oscillator networks with SN-symmetry, we look for non-degenerate Hopf bifurcations as the time-delay between nodes increases. The remarkably simple stability criterion for synchronous solutions which, in the case of first-order self-oscillators, states that stability depends only on the sign of the coupling function derivative, is extended to a generic coupling function for second-order oscillators. As an application example, the stability boundaries for a N-node Phase-Locked Loop network is analysed.