Singularly perturbed boundary-focus bifurcations

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