  As parameters are varied a boundary equilibrium bifurcation (BEB) occurs when
an equilibrium collides with a discontinuity surface in a piecewise-smooth
system of ODEs. Under certain genericity conditions, at a BEB the equilibrium
either transitions to a pseudo-equilibrium (on the discontinuity surface) or
collides and annihilates with a coexisting pseudo-equilibrium. These two
scenarios are distinguished by the sign of a certain inner product. Here it is
shown that this sign can be determined from the number of unstable directions
associated with the two equilibria by using techniques developed by Feigin. A
new normal form is proposed for BEBs in systems of any number of dimensions.
The normal form involves a companion matrix, as does the leading order sliding
dynamics, and so the connection to the stability of the equilibria is explicit.
In two dimensions the parameters of the normal form distinguish, in a simple
way, the eight topologically distinct cases for the generic local dynamics at a
BEB. A numerical exploration in three dimensions reveals that BEBs can create
multiple attractors and chaotic attractors, and that the equilibrium at the BEB
can be unstable even if both equilibria are stable. The developments presented
here stem from seemingly unutilised similarities between BEBs in discontinuous
systems (specifically Filippov systems as studied here) and BEBs in continuous
systems for which analogous results are, to date, more advanced.
