TY - UNPB

T1 - The singular limit of a Hopf bifurcation

AU - Guckenheimer, J

AU - Osinga, Hinke M

N1 - Sponsorship: The research of H.M.O. was supported by an EPSRC Advanced Research Fellowship grant. The research of J.G. was partially supported by grants from the National Science Foundation and the Department of Energy.

PY - 2011/6

Y1 - 2011/6

N2 - 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 slow-fast system persist in the fast subsystem obtained from this singular limit. However, the Hopf bifurcation of the fast subsystem does not necessarily have the same criticality as the corresponding Hopf bifurcation of the full slow-fast 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 slow-fast 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 fast subsystem in the limit as epsilon goes to 0; furthermore, the stability of the orbits is dictated by that of the fast subsystem. 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.

AB - 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 slow-fast system persist in the fast subsystem obtained from this singular limit. However, the Hopf bifurcation of the fast subsystem does not necessarily have the same criticality as the corresponding Hopf bifurcation of the full slow-fast 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 slow-fast 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 fast subsystem in the limit as epsilon goes to 0; furthermore, the stability of the orbits is dictated by that of the fast subsystem. 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.

M3 - Working paper and Preprints

BT - The singular limit of a Hopf bifurcation

ER -