In the proposed chapter, we are going to outline the results of a study on an arithmetical plane of a broad family of dynamic systems having polynomial right parts. Let these polynomials be of cubic and square reciprocal forms. The task of our investigation is to find out all the different (in the topological sense) phase portraits in a Poincare circle and indicate the coefficient criteria of their appearance. To achieve this goal, we use the Poincare method of central and orthogonal consecutive displays (or mappings). As a result of this thorough investigation, we have constructed more than 250 topologically different phase portraits in total. Every portrait we present using a special table called a descriptive phase portrait. Each line of such a special table corresponds to one invariant cell of the phase portrait and describes its boundaries, as well as a source of its phase flow and a sink of it.
Part of the book: Differential Equations