Oscillator death in coupled functional differential equations near Hopf bifurcation

The stability of the equilibrium solution is analyzed for coupled systems of retarded functional differential equations near a supercritical Hopf bifurcation. Necessary and sufficient conditions are derived for asymptotic stability under general coupling conditions. It is shown that the largest eigenvalue of the graph Laplacian completely characterizes the effect of the connection topology on the stability of diffusively and symmetrically coupled identical systems. In particular, all bipartite graphs have identical stability characteristics regardless of their size. Furthermore, bipartite graphs and large complete graphs provide, respectively, lower and upper bounds for the parametric stability regions for arbitrary connection topologies. Generalizations are given for networks with asymmetric coupling. The results characterize the connection topology as a mechanism for the death of coupled oscillators near Hopf bifurcation.