The singular limit of a Hopf bifurcation
John Guckenheimer  Mathematics Department, Cornell University, Ithaca, NY 14853, United States (email) Abstract: Hopf bifurcation in systems with multiple time scales takes several forms, depending upon whether the bifurcation occurs in fast directions, slow directions or a mixture of these two. Hopf bifurcation in fast directions is influenced by the singular limit of the fast time scale, that is, when the ratio $\epsilon$ of the slowest and fastest time scales goes to zero. The bifurcations of the full slowfast system persist in the layer equations obtained from this singular limit. However, the Hopf bifurcation of the layer equations does not necessarily have the same criticality as the corresponding Hopf bifurcation of the full slowfast system, even in the limit $\epsilon \to 0$ when the two bifurcations occur at the same point. We investigate this situation by presenting a simple slowfast system that is amenable to a complete analysis of its bifurcation diagram. In this model, the family of periodic orbits that emanates from the Hopf bifurcation accumulates onto the corresponding family of the layer equations in the limit as $\epsilon \to 0$; furthermore, the stability of the orbits is dictated by that of the layer equation. We prove that a torus bifurcation occurs $O(\epsilon)$ near the Hopf bifurcation of the full system when the criticality of the two Hopf bifurcations is different.
Keywords: Hopf bifurcation, slowfast system, Lyapunov coefficient, torus bifurcation
Received: June 2011; Revised: November 2011; Available Online: March 2012. 
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