Stability switches, bifurcation, and multi-stability of coupled networks with time delays

This paper reveals the dynamical behaviors of a network consisting of a pair of ring sub-networks and two-way couplings of neurons between the individual sub-networks. Time delays are introduced into the internal connections within the individual sub-networks and the couplings between the individual sub-networks. The stability and instability of the trivial equilibrium of the network are analyzed and the conditions for the existence of Hopf bifurcation are obtained by discussing the associated characteristic equation. The criterion for the global stability of the trivial equilibrium of the network is given by constructing a suitable Lyapunov functional. Numerical simulations are performed to validate the theoretical results and rich dynamical behaviors are observed, such as multiple stability switches of the network equilibrium, synchronous/asynchronous oscillations, and multi-stability.