On nonlocal bifurcations in two-parameter families of vector fields on the plane with involutive symmetry

Abstract

Background. The study of dynamical systems that are invariant with respect to different groups of transformations is important both for the theory of differential equations and for its applications. Local bifurcations in generic two-parameter families of dynamical systems defined by vector fields invariant under the involution of a plane having a line of fixed points were described by H. Zholondek. The purpose of this research is to study some nonlocal bifurcations in such families. Materials and methods. The method of point mappings and other methods of the qualitative theory of differential equations are applied. Results. We consider a generic two-parameter family of planar vector fields with symmetry about the x-axis. It is assumed that at a zero value of the parameter, the field has a rough saddle, a weak saddle lying on the x axis, and two symmetrical contours formed by the separatrices of these saddles. A bifurcation diagram is obtained – a partition of the neighborhood of zero on the parameter plane by types of phase portraits in the neighborhood of a polycycle composed of these contours. In particular, we show that one stable rough limit cycle can be born from each contour. Conclusions. One of the possible scenarios for the occurrence of stable periodic oscillations when changing the parameters of a dynamical system with involutive symmetry is described.

Keywords

• planar vector field
• dynamical system
• saddle
• weak saddle
• separatrix
• limit cycle
• bifurcation diagram