The singular limit of a Hopf bifurcation

John Guckenheimer; Hinke M. Osinga

doi:10.3934/dcds.2012.32.2805

Article Contents

2012, Volume 32, Issue 8: 2805-2823. Doi: 10.3934/dcds.2012.32.2805

This issue Previous Article Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold Next Article How to find a codimension-one heteroclinic cycle between two periodic orbits

The singular limit of a Hopf bifurcation

John Guckenheimer^1, and
Hinke M. Osinga^2,

1.
Mathematics Department, Cornell University, Ithaca, NY 14853
2.
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142

Received: May 31, 2011

Revised: October 31, 2011

Published: July 2012

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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 slow-fast 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 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 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.

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Mathematics Subject Classification: 34C20, 34D15, 37G10.

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