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

TL;DR

This paper investigates the stability of periodic solutions near Hopf bifurcations in time-delayed fully-connected PLL networks, using center manifold reduction and numerical simulations to extend understanding of network dynamics.

Contribution

It provides a detailed analysis of Hopf bifurcation stability in delayed PLL networks, extending previous work to include stability near bifurcation points and confirming results with simulations.

Findings

Hopf bifurcations lead to stable and unstable periodic solutions

Center manifold reduction simplifies the infinite-dimensional problem

Numerical simulations confirm analytical stability results

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 the nature in the solutions in the parameter space for a network of Phase-Locked Loop oscillators was studied in Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators. Communications in Nonlinear Science and Numerical Simulation, Elsevier BV, 2014, (on-line version), 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 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.