Bifurcation of periodic orbits with multiple crossings in a class of planar Filippov systems

In this paper we discuss bifurcation of periodic orbits of planar Filippov systems with one switching line and the unperturbed periodic orbits cross the switching line transversally multiple times. When the unperturbed system has a limit cycle, we give a condition for its persistence; when the unperturbed system has an annulus of periodic orbits, we derive an expression of the first order Melnikov function which can be used to study the number of limit cycles bifurcated from the annulus. As an application, we construct a piecewise cubic system with 44 limit cycles bifurcated from the periodic annulus.