Bifurcations of a polycycle formed by separatrices of a saddle with zero saddle value of a dynamical system with central symmetry

Abstract

We consider two-parameter families of planar vector fields with central symmetry. Assume that for zero values of the parameters, the field has a hyperbolic saddle at the origin O and two symmetric loops of the separatrices of this saddle. The saddle value – the trace of the matrix of the linear part of the field at the point O – is assumed to be zero. We describe the bifurcation diagram of a generic family – a partition of a neighborhood of the origin on the parameter plane into topological equivalence classes of dynamical systems defined by these vector fields in a fixed neighborhood U of the polycycle formed by loops of separatrices. In particular, for each element of the partition, the number and type of the field belonging to U are indicated.