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We establish a theorem on bifurcation of limit cycles from a focus boundaryequilibrium of an impacting system, which is universally applicable to provebifurcation of limit cycles from focus boundary equilibria in other types ofpiecewise-smooth systems, such as Filippov systems and sweeping processes.Specifically, we assume that one of the subsystems of the piecewise-smoothsystem under consideration admits a focus equilibrium that lie on the switchingmanifold 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 occurrenceof a finite-time stable limit cycle from the above-mentioned focus equilibriumwhen the parameter crosses the bifurcation value. Examples illustrate howconditions of our theorems lead to closed-form formulas for the coefficients ofthe linearized system.

Bifurcations of finite-time stable limit cycles from focus boundary equilibria in impacting systems, Filippov systems and sweeping processes