Global Analysis on the Discontinuous Limit Case of a Smooth Oscillator

Global dynamics of a class of planar Filippov systems with symmetry, which is a discontinuous limit case of a smooth oscillator, is studied. Necessary and sufficient conditions for the existence and the number of limit cycles are given. It is shown that at most two limit cycles or a pair of grazing loops exist. A special method is introduced to study grazing bifurcation. The monotonicity and the C-infinity smoothness of the grazing bifurcation curve are proved. All global phase portraits and a complete global bifurcation diagram are described. Finally, some numerical examples are demonstrated.