Investigation of a Family of Dynamic Systems with Reciprocal Polynomial Right Parts in a Poincare Circle

Abstract

Abstract A paper describes methods and results of a fundamental study of some family of dynamic systems having reciprocal polynomial right parts, which is considered on the arithmetical (real) plane. One of the equations in these systems includes a cubic form in its right part, while the other one includes a square form. The goal was to find out all topologically different phase portraits possible for differential dynamic systems under consideration in a Poincare circle and outline close to coefficient criteria of them. A Poincare method of consecutive central and orthogonal mappings has been applied, and allowed to obtain more than 230 independent phase portraits. Each phase portrait has been described with a special table, every line of which corresponds to one invariant cell of the portrait and describes its boundary, as well as a source and a sink of its phase flow. All finite and infinitely remote singularities of considered dynamic systems were investigated.