Stability Changes of Periodic Solutions to a Coupled Nonlinear Equation with Time Delay

Abstract

We shall consider a linear difference coupled equation with time delay and discuss a Hopf bifurcation of this equation. In the case of weakly coupling, it will be shown that two types of periodic solutions bifurcate from the steady state for some parameter values, and that those periodic solutions exchange the stability under certain conditions; moreover, under another conditions one of those periodic solutions changes its stability twice at least. Sufficient conditions for the occurrence of such phenomena will be presented along with specific examples.