Bifurcations of a polycycle formed by two separatrix loops of a non-rough saddle of a dynamical system with central symmetry

A family of smooth dynamical systems defined on the plane and depending on a two-dimensional parameter varying in a neighborhood of zero is considered. All systems of the family are assumed to be invariant under a symmetry transformation about the origin. At zero value of the parameter, the dynamical system has the simplest non-rough saddle, both outgoing separatrixes of which go to the same saddle, forming two loops. The polycycle “eight”, consisting of the loops, is an attractor of this system. It has a neighborhood $U$, at the boundary points of which all trajectories of systems of the family with parameters close to zero enter $U$. Under the condition of general position, bifurcations in the neighborhood $U$ of the polycycle are described when the parameter changes. The values of the parameter in a small neighborhood of zero, for which the system is non-rough in $U$, form five smooth curves entering the origin, dividing this neighborhood into connected components, for values of the parameter from which the systems of the family are rough. For each component, the topological type of the corresponding dynamical systems in $U$ is described. In particular, the regions of the parameter are indicated for which the system has a single attractor in $U$ — a node, two attractors — a node and a cycle homotopic in $U$ to a polycycle, or two symmetric cycles homotopic in $U$ to loops from a polycycle, as well as three attractors — a node and two symmetric cycles.