Piecewise smooth dynamical systems: Persistence of periodic solutions and normal forms

We consider an n  -dimensional piecewise smooth vector field with two zones separated by a hyperplane Σ which admits an invariant hyperplane Ω transversal to Σ containing a period annulus AA fulfilled by crossing periodic solutions. For small discontinuous perturbations of these systems we develop a Melnikov-like function to control the persistence of periodic solutions contained in AA. When n=3n=3 we provide normal forms for the piecewise linear case. Finally we apply the Melnikov-like function to study discontinuous perturbations of the given normal forms.