Infinitely Remote Singularities of Special Differential Dynamic Systems

Abstract

The work is devoted to the results of a fundamental study on thearithmetical plane of a broad special family of differential dynamic systemshaving polynomial right parts. Let those polynomials be a cubic and a squarereciprocal forms. A task of a whole investigation was to find out alltopologically different phase portraits in a Poincare circle and indicate closeto coefficient criteria of them. To achieve this goal a Poincare method of thecentral and the orthogonal consecutive displays (or mappings) has been used. Asa rezult more than 250 topologically different phase portraitsin a total have been constructed. Every portrait we depict with a special tablecalled a descriptive phase portrait. Each line of such a special tablecorresponds to one invariant cell of the phase portrait and describes itsboundary, a source of its phase flow and a sink of it. All finite andinfinitely remote singularities of dynamic systems under consideration werefully investigated. Namely infinitely remote singularities are discussed in thepresent article.