Continuation of the Periodic Orbits for the Differential Equation with Discontinuous Right Hand Side

In this paper the discontinuous system with one parameter perturbation is considered. Here the unperturbed system is supposed to have at least either one periodic orbit or a limit cycle. The aim is to prove the continuation of the periodic orbits under perturbation by means of the bifurcation map and the zeroes of this map imply the periodic orbits for the perturbed system. The tools for this problem are jumps of fundamental matrix solutions and the Poincare map for discontinuous systems. Therefore, we develop the Diliberto theorem and variation lemma for the system with discontinuous right hand side. At the end, as application of our method, the effect of discontinuous damping on Van der pol equation, and the effect of small force on the discontinuous linear oscillator with add a ·sgn(x) are considered.