On the separatris loop bifurcations of two-dimensional piecewise-smooth dynamic system

Background. Bifurcations in generic one- and two-parameter families of smooth dynamical systems on the plane are almost completely studied. For applications, piecewise-smooth dynamical systems in the plane are of considerable interest. There are much more different types of bifurcations for them than for smooth dynamical systems. Some of them are already described. However, the continuation of the study of bifurcations in generic two-parameter families of two-dimensional piecewise-smooth dynamical systems seems to be still relevant.
Materials and methods. We use methods of the qualitative theory of differential equations.
Results. We consider a two-dimensional piecewise smooth vector field X. Let S be a point on the line of discontinuity of the field, and in its semi-neighborhoods V1 and V2 the field coincides with smooth vector fields, respectively, Х1 and Х2. For the field Х1, the point S is a saddle with nonzero saddle value, whose invariant manifolds are transversal to the line of discontinuity. At the point S vector field Х2 is transversal to the line of discontinuity and directed inwards V1. The outgoing and incoming separatrixes of the saddle S that start at V1 do not contain singular points and form a loop together with S. For generic two-parameter deformations of the considered vector fields in the neighborhood of the loop, bifurcation diagrams are obtained.
Conclusions. Bifurcations of the separatrix loop of singular point on the line of discontinuity of the vector field are described.