On the bifurcations of two periodic trajectories of a piecewise-smooth dynamical system with central symmetry

Abstract

Background. A large number of scientific works are devoted to the description of bifurcations in generic families of piecewise-smooth dynamical systems on the plane. Although dynamical systems with symmetry are often encountered in applied problems, the bifurcations of piecewise-smooth systems with symmetry have been studied not enough. Therefore, the consideration of bifurcations in generic families of such dynamical systems is of undoubted interest. Materials and methods. The methods of the qualitative theory of differential equations are applied. The behavior of the Poincare mappings and the corresponding divergence functions for different values of the parameters is investigated. We used estimates for the derivatives of local correspondence functions along trajectories at the points of tangency of the trajectories with the line of discontinuity of the vector field. Results. We consider a piecewise-smooth vector field X on a plane, “sewn” from smooth vector fields specified in the upper and lower half-planes, respectively, and having periodic trajectories tangent to the x-axis, which is invariant under symmetry transformation about the origin. The bouquet Г, composed of the indicated periodic trajectories, is a periodic trajectory of the field X. For a two-parameter family in general position, which is a deformation of the field X in the space of piecewise-smooth vector fields with central symmetry, bifurcations are described in a neighborhood U of the contour Г. The bifurcation diagram is obtained – a partition of a neighborhood of zero in the parameter planes into topological equivalence classes in U of vector fields of the family. Conclusions. Generic two-parameter bifurcations in a neighborhood of the considered bouquet of periodic trajectories are described.

Keywords

• piecewise smooth vector field
• symmetry
• periodic trajectory
• bifurcation
• bifurcation diagram