A theoretical justification of the method of harmonic balance for systems with discontinuities

We prove a theorem which provides a rigorous justification of an intuitive method used by electrical engineers to predict the presence or absence of periodic oscillations in nonlinear systems. Although the literature contains some excellent discussions of the conditions under which the method can be rigorously justified, there are some oversights and there is lacking a completely detailed treatment, particularly for unforced discontinuous systems. By applying the theory of topological degree for differential inclusions, we are able to present a unified rigorous justification in full detail, and we can illustrate how our abstract hypotheses match up, point by point, with the standard hypotheses used by engineers.