On bifurcations of a periodic trajectory “eight” of a piesewise smooth vector field with symmetry

Background. The study of bifurcation in typical one- and two-parameter families of piecewise smooth dynamical systems on the plane is of considerable interest, both from a theoretical and applied point of view. A large number of scientific papers are devoted to these studies. In applications, dynamic systems with symmetry are often found. However, bifurcations of piecewise smooth systems with symmetry have so far been little studied. Therefore, the study of bifurcations in typical families of such dynamical systems seems relevant.
Materials and methods. We use methods of the qualitative theory of differential equations. The main method is to study the behavior of Poincare mappings and the corresponding divergence functions for various parameter values.
Results. We consider a two-parameter family of piecewise-smooth vector fields in the plane that are “stitched” from smooth vector fields defined respectively in the upper and lower half-planes. The vector fields of the family are assumed to be invariant at the transformation of symmetry with respect to the origin. At zero values of the parameters, the vector field has an orbital stable periodic trajectory $\Gamma$, homeomorphic to the “eight”, tangent to the x axis at the origin and above and below. In the generic case, bifurcations are described in a neighborhood $U$ of the contour $\Gamma$. A bifurcation diagram is obtained - a partition of a neighborhood of zero on the parameter plane into classes of topological equivalence in $U$ of vector fields of the family.
Conclusions. Generic two-parameter bifurcations in a neighborhood of the considered periodic trajectory are described.