On Bifurcations of Certain Separatrix Contours of a Piecewise-Smooth Dynamical System with Symmetry

This article considers a one-parameter family of piecewise-smooth vector fields that are invariant under reflection from the x-axis on a plane with Cartesian coordinates (x, y). The switching line passes through the origin O, transversally to the x-axis. For a zero value of the parameter, let the vector field of the family in the left half-neighborhood of the switching line coincide with a smooth vector field that has the O point as a rough stable node, and in its right half-neighborhood it coincides with a smooth vector field without singular points. Let this field also have a rough saddle S on the x-axis such that the open arc of the x-axis between the O and S points is an incoming separatrix of the saddle, and the two symmetric outgoing separatrices of the saddle do not contain any singular points and lead to the O point. The article demonstrates that if there is no singular point in the left semi-neighborhood of the switching line for the positive values of the parameter, then a unique, stable, periodic trajectory arises from each of the two symmetrical contours formed by the separatrices. Under certain additional conditions, the emerging periodic trajectory is unique and hyperbolic.