Singularly perturbed boundary-equilibrium bifurcations

Boundary equilibria bifurcation (BEB) arises in piecewise-smooth (PWS) systems when an equilibrium collides with a discontinuity set under parameter variation.Singularly perturbed BEB refers to a bifurcation arising in singular perturbation problems which limit as some → 0 to PWS systems which undergo a BEB.This work completes a classification for codimension-1 singularly perturbed BEB in the plane initiated by the present authors in [19], using a combination of tools from PWS theory, geometric singular perturbation theory and a method of geometric desingularization known as blow-up.After deriving a local normal form capable of generating all 12 singularly perturbed BEBs, we describe the unfolding in each case.Detailed quantitative results on saddle-node, Andronov-Hopf, homoclinic and codimension-2 Bogdanov-Takens bifurcations involved in the unfoldings and classification are presented.Each bifurcation is singular in the sense that it occurs within a domain which shrinks to zero as → 0 at a rate determined by the rate at which the system loses smoothness.Detailed asymptotics for a distinguished homoclinic connection which forms the boundary between two singularly perturbed BEBs in parameter space are also given.