Singularly Perturbed Boundary-Focus Bifurcations

TL;DR

This paper analyzes how boundary-focus bifurcations in piecewise-smooth systems influence the dynamics of smooth systems as the smoothing parameter approaches zero, revealing bifurcation structures and cycle persistence.

Contribution

It introduces a local normal form for boundary-focus bifurcations and characterizes the bifurcation structures and cycle persistence in smooth systems near the PWS limit.

Findings

Identification of supercritical Andronov-Hopf bifurcation in one case.

Detection of supercritical Bogdanov-Takens bifurcation in other cases.

Persistence of PWS cycles as relaxation cycles in smooth systems.

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.