Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems

This paper deals with the problem of periodic orbit bifurcations for high-dimensional piecewise smooth systems. Under the assumption that the unperturbed system has a family of periodic orbits which are transversal to the switch plane, a formula for the first order Melnikov vector function is developed which can be used to study the number of periodic orbits bifurcated from the periodic orbits. We especially can use the function to study the number of periodic orbits both in degenerate Hopf bifurcations and in degenerate homoclinic bifurcations. Finally, we present two examples to illustrate an application of the theoretical results.