The Persistence of Periodic Orbits in a Class of Three-Dimensional Piecewise Smooth Systems

This work explores the persistence of periodic orbits under small discontinuous perturbations to a three-dimensional piecewise smooth vector field. We assume that the unperturbed system has an invariant hyperplane that intersects the switching plane transversally and contains a family of crossing periodic orbits around the origin. By developing the Poincaré map and defining the displacement function, we derive some sufficient conditions for the existence of periodic solutions. Furthermore, we apply these results to specific examples where the origin is either a regular two-fold singularity or a 1-degenerate two-fold singularity, and obtain the parameter conditions for the existence of four or six crossing limit cycles.