Bifurcations in time-delay fully-connected networks with symmetry

Abstract

In this work a brief method for finding steady-state and Hopf bifurcations in a (R + 1)-th order N-node time-delay fully-connected network with symmetry is explored. A self-sustained Phase-Locked Loop is used as node. The irreducible representations found due to the network symmetry are used to find regions of time-delay independent stability/instability in the parameter space. Symmetry-preserving and symmetry-breaking bifurcations can be computed numerically using the Sn map proposed in [1]. The analytic results show the existence of symmetry-breaking bifurcations with multiplicity N − 1. A second-order N-node network is used as application example. This work is a generalization of some results presented in [2].