On bifurcations of a periodic orbit tangent to switching lines at two points

Background. Dynamic systems defined by discontinuous piecewise smooth vector fields on a plane are natural mathematical models of relay systems in automatic control theory. Periodic trajectories describe self-oscillations. Although a significant number of works have been devoted to the study of the birth of periodic trajectories, the description of typical bifurcations is far from complete. The purpose of this research is to study bifurcations of periodic trajectories similar to bifurcations of double and triple cycles of a smooth dynamic system. Materials and methods. The method of point mappings and other methods of the qualitative theory of differential equations are applied. Results. A generic two-parameter family of piecewise smooth vector fields on a plane is considered. It is assumed that for zero values of the parameters the field has a periodic trajectory Г touching the switching lines at two singular points of the fork type and not containing other singular points. In this case, both components into which Г divides the plane intersect with the separatrices of the forks that are not contained in Г. Three cases are considered. In the first case, Г is stable and bifurcates similarly to a triple cycle, in the second case, Г is stable, but its bifurcations consist only in changing the number of sections of sliding motions on it, and in the third case, Г is semistable and bifurcates similarly to a double cycle. Conclusions. Several possible scenarios for the birth and rebirth of periodic trajectories of a piecewise smooth dynamic system when its parameters change are indicated.