Delayed Hopf bifurcation in time-delayed slow-fast systems

This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems. Here the two delayed's have different meanings. The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point, but at some other point which is above the bifurcation point by an obvious distance. In a time-delayed system, the evolution of the system depends not only on the present state but also on past states. In this paper, the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction, and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory. It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems, and the theoretical prediction on the exit-point is in good agreement with the numerical calculation, as illustrated in the two illustrative examples.