  We establish a theorem on bifurcation of limit cycles from a focus boundary
equilibrium of an impacting system, which is universally applicable to prove
bifurcation of limit cycles from focus boundary equilibria in other types of
piecewise-smooth systems, such as Filippov systems and sweeping processes.
Specifically, we assume that one of the subsystems of the piecewise-smooth
system under consideration admits a focus equilibrium that lie on the switching
manifold at the bifurcation value of the parameter. In each of the three cases,
we derive a linearized system which is capable to conclude about the occurrence
of a finite-time stable limit cycle from the above-mentioned focus equilibrium
when the parameter crosses the bifurcation value. Examples illustrate how
conditions of our theorems lead to closed-form formulas for the coefficients of
the linearized system.
