Stability of Hopf bifurcations in Time-delayed fully-connected PLL networks

TL;DR

This paper investigates the stability of small-amplitude periodic solutions near Hopf bifurcations in time-delayed fully-connected PLL networks, extending previous bifurcation analysis with numerical validation.

Contribution

It provides a detailed stability analysis of Hopf bifurcations in delayed PLL networks using center manifold reduction, with numerical confirmation of theoretical results.

Findings

Stability conditions for periodic solutions near Hopf bifurcations

Numerical simulations confirming analytical predictions

Extension of results to higher-order oscillator networks

Abstract

Dynamics in delayed differential equations (DDEs) is a well studied problem mainly because DDEs arise in models in many areas of science including biology, physiology, population dynamics and engineering. The change of nature in the solutions in the parameter space for a network of Phase-Locked Loop oscillators was studied in \textit{Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators}. Communications in Nonlinear Science and Numerical Simulation, Elsevier BV, Volume 22, Issues 1-3, May 2015, Pages 793-820, where the existence of Hopf bifurcations for both cases, symmetry-preserving and symmetry-breaking synchronization was well stablished. In this work we continue the analysis exploring the stability of small-amplitude periodic solutions emerging near Hopf bifurcations in the Fixed-point subspace, based on the reduction of the infinite-dimensional space onto a two-dimensional center manifold. Numerical simulations are presented in order to confirm our analitycal results. Although we explore network dynamics of second-order oscillators, results are extendable to higher order nodes.