Hopf bifurcation and spatio-temporal patterns in a hierarchical network with delays and Z2×Zn symmetry

A hierarchical network composed of two interacting rings each of which consists of n identical cells with an unidirectional coupling is the topic of this paper. We present a detailed discussion about the linear stability of the equilibrium by analyzing the associated characteristic equation. The local Hopf bifurcation and spatio-temporal patterns of bifurcating periodic oscillations are also given by employing the symmetric Hopf bifurcation theory for delay differential equations. In particular, by using the normal form theory and the center manifold theorem, we derive the formula determining the direction of the Hopf bifurcation and the stability of the bifurcated periodic orbits. An example with numerical simulations is presented to illustrate our theoretical results.