Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems

Abstract

We first study the problem of chaotic behaviour in time-perturbed discontinuous systems whose unperturbed part has a piecewise C1 homoclinic solution transversally crossing the discontinuity manifolds. We show that if a certain Melnikov function has a simple zero at some point, then the system has solutions that behave chaotically. The Melnikov function is explicitly constructed for two-dimensional systems and extends the usual Melnikov function for the smooth case. In the second part, we extend these results to sliding homoclinic bifurcations. We also mention some possibilities for further research.