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 epsilon -> 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 epsilon -> 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. Finally, we describe the explosive onset of oscillations arising in the unfolding of a particular singularly perturbed boundary-node bifurcation. We prove the existence of the oscillations as perturbations of PWS cycles, and derive a growth rate which is polynomial in epsilon and dependent on the rate at which the system loses smoothness. For all the results presented herein, corresponding results for regularised PWS systems are obtained via the limit epsilon -> 0.

Automated summary

This work completes a classification of codimension-1 singularly perturbed boundary equilibria bifurcation (BEB) in the plane, utilizing tools from PWS theory, geometric singular perturbation theory, and the method of geometric desingularization known as blow-up. Local normal forms for generating all 12 singularly perturbed BEBs are derived, and the unfolding in each case is described. Detailed quantitative results on various bifurcations involved in the unfoldings and classification are presented, including saddle-node, Andronov-Hopf, homoclinic, and codimension-2 Bogdanov-Takens bifurcations.