Qualitative Investigation of Some Hierarchical Family of Cubic Dynamic Systems

The article represents methods and results of the original investigation of some hierarchical family of cubic dynamic differential systems in the Poincaré disk. For a family of systems, whose right-hand sides are reciprocal polynomial forms of the phase variables (a cubic form in the first equation and a square form in another one) all possible topologically different phase portraits in the Poincaré disk were investigated and constructed using approaches of the qualitative theory of ODEs and dynamic systems. Close to coefficient criteria of all phase portraits appearance for each of the existing subfamilies of systems under consideration were obtained. Research methods invented especially for the goals of this study are described in the paper.