Bifurcations of a sewn focus of a piecewise smooth dynamical system with central symmetry

In this paper, we study planar dynamical systems, given by piecewise smooth vector fields depending on parameters. The generation of periodic trajectories from a singular point on the field discontinuity line when changing parameters under different conditions has been considered in many works. In particular, the bifurcations of the sewn (fused) focus, analogous to the Andronov-Hopf bifurcation of the composed focus of a smooth vector field, were studied. Since dynamical systems used in applications often possess various kinds of symmetry, the study of bifurcations in such systems is of undoubted interest. We consider a piecewise smooth vector field "sewn" from smooth vector fields defined in the upper and lower half-planes, which does not change under the symmetry transformation for the origin of coordinates O and has a sewn focus of multiplicity one or two at the origin of coordinates. We describe bifurcations of phase portraits in a neighborhood of the point O, respectively, under generic one-parameter and two-parameter perturbations of the vector field. In particular, the domains of parameters for which limit cycles exist in a neighborhood of O are indicated.