Periodic Orbits in a Second-Order Discontinuous System with an Elliptic Boundary

This paper develops the analytical conditions for the onset and disappearance of motion passability and sliding along an elliptic boundary in a second-order discontinuous system. A periodically forced system, described by two different linear subsystems, is considered mainly to demonstrate the methodology. The passable, sliding and grazing conditions of a flow to the elliptic boundary in the discontinuous dynamical system are provided through the analysis of the corresponding vector fields and G-functions. Moreover, by constructing appropriate generic mappings, periodic orbits in such a discontinuous system are predicted analytically. Finally, three different cases are discussed to illustrate the existence of periodic orbits with passable and/or sliding flows. The results obtained in this paper can be applied to the sliding mode control in discontinuous dynamical systems.