Stability changes of periodic solutions to a coupled nonlinear equation with time delay (Q1065998)

A linear difference coupled equation is considered and its Hopf bifurcation is discussed. In the case of weak coupling, it is shown that two types of periodic solutions bifurcate from the steady state for some parameter values, and that they exchange the stability under a certain condition while one of them changes its stability twice under another condition. Sufficient conditions for the occurrence of such phenomena are presented along with specific examples. A further study [the author, ''Secondary bifurcating periodic solutions to a two coupled oscillators with time delay'', to appear in Jap. J. Appl. Math.] shows a complete structure of the secondary bifurcation of periodic solution around a singularity for a class of weakly coupled equations which include the above. By this result a striking deformation of the branch of secondary bifurcation is also observed as the coupling term is varied in some way.