On bifurcation of separatrix contours of planar vector fieds with involutive symmetry

We consider a two-parameter family of vector fields in the plane with symmetry about the x-axis. It is assumed that at zero values of the parameters, the vector field has a saddle-node with a negative eigenvalue of the matrix of the linear part of the field and a rough saddle lying on the x-axis, as well as two symmetric contours formed by the outgoing separatrices of the saddle, coinciding with the incoming separatrices of the saddle-node. A bifurcation diagram of such a family is described – a partition of the neighborhood of zero on the parameter plane by types of phase portraits in a neighborhood of the union of these contours. In particular, it is shown that one stable rough limit cycle can be born from each contour.