Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations |
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Authors: | R Huzak P De Maesschalck F Dumortier |
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Institution: | Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek, Belgium |
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Abstract: | This paper deals with local bifurcations occurring near singular points of planar slow-fast systems. In particular, it is concerned with the study of the slow-fast variant of the unfolding of a codimension 3 nilpotent singularity. The slow-fast variant of a codimension 1 Hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the Van der Pol system. Similarly, codimension 2 slow-fast Bogdanov–Takens bifurcations have been characterized. Here, the singularity is of codimension 3 and we distinguish slow-fast elliptic and slow-fast saddle bifurcations. We focus our study on the appearance on small-amplitude limit cycles, and rely on techniques from geometric singular perturbation theory and blow-up. |
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Keywords: | 34E15 37G15 34E20 34C23 34C26 |
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