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arXiv · 2006.06087

Singularly Perturbed Boundary-Focus Bifurcations

Samuel Jelbart·Kristian Uldall Kristiansen·Martin Wechselberger
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Abstract

We consider smooth systems limiting as $\epsilon \to 0$ to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with $0 < \epsilon \ll 1$ using a combination of geometric singular perturbation theory and blow-up. We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an $\epsilon-$dependent domain which shrinks to zero as $\epsilon \to 0$, identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation cycles in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation cycles to regular cycles within the $\epsilon-$dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction.

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Samuel Jelbart, Kristian Uldall Kristiansen, Martin Wechselberger. 2020-06-10. Singularly Perturbed Boundary-Focus Bifurcations. https://arxiv.org/abs/2006.06087

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