Singularly perturbed boundary-focus bifurcations

We consider smooth systems limiting as epsilon -> 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 smoothsystem with sufficiently small but non-zero epsilon, using a combination of geometric singular perturbation theoryand 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 -> 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 oscillations in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation oscillations 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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

We study the transition of smooth systems to piecewise-smooth systems with a boundary-focus bifurcation as epsilon -> 0, and identify different bifurcation structures. We uncover the evolution characteristics of cycles associated with BF bifurcations in the smooth system, and prove the existence of a family of stable limit cycles.