Abstract
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 ϵ → 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 [], 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 ϵ → 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 ϵ 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 ϵ → 0.
Highlights
This manuscript concerns the unfolding of singularities in planar singular perturbation problems which limit to piecewise-smooth (PWS) systems
After deriving a local normal form capable of generating all 12 singularly perturbed Boundary equilibria bifurcation (BEB), we describe the unfolding in each case
BEs are PWS singularities which unfold generically in a codimension-1 bifurcation known as a boundary equilibrium bifurcation (BEB), whereby an isolated equilibrium collides with the switching manifold Σ under parameter variation
Summary
This manuscript concerns the unfolding of singularities in planar singular perturbation problems which limit to piecewise-smooth (PWS) systems. It is natural to consider a class of smooth singular perturbation problems, which limit to PWS systems that are discontinuous along a switching manifold Σ as a perturbation parameter → 0. To [17, 19], emphasis is placed on understanding the smooth dynamics with 0 < 1 This allows for the treatment of problems arising either naturally or via regularization simultaneously, since the corresponding results for (regularized) PWS systems are obtained upon taking the non-smooth singular limit → 0. It is worthy to note that within the class of smooth monotonic regularizations considered, the dynamics are shown to be qualitatively determined by the underlying PWS problem, i.e. the bifurcation structure is qualitatively independent of the choice of regularization, and determined by the type of PWS unfolding in the limit → 0.
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