  Boundary equilibria bifurcation (BEB) arises in piecewise-smooth 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 $\epsilon \to 0$ to piecewise-smooth
(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 (GSPT) 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 $\epsilon \to 0$ 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 (BN) 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 regularized PWS systems are obtained via the
limit $\epsilon \to 0$.
