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 arising singular perturbation problems which limit as some ? 0 PWS undergo BEB. This work completes classification for codimension-1 singularly the plane initiated by present authors [19], using combination of tools from theory, geometric theory and method desingularization known blow-up. After deriving local normal form capable generating all 12 BEBs, we describe unfolding each case. Detailed quantitative results on saddle-node, Andronov–Hopf, homoclinic codimension-2 Bogdanov–Takens bifurcations involved unfoldings are presented. Each is sense that it occurs within domain shrinks zero at rate determined system loses smoothness. asymptotics distinguished connection forms boundary between two BEBs space also given. Finally, explosive onset oscillations particular boundary-node bifurcation. We prove existence perturbations cycles, derive growth polynomial dependent For presented herein, corresponding regularised obtained via 0.