Investigation of Phase Portraits Belonging to Polynomial Dynamic Systems in a Poincare Disk

The paper describes especially developed research methods and results for some original study of a broad dynamic systems' family, which is characterized with the reciprocal polynomials in the right parts of differential equations which compose a system. The goal of this work is to obtain and analyze all the existing (and different in the topological meaning) phase portraits in a Poincare disk, as well as to indicate the criteria of each topological type of phase portraits appearance. Poincare methods of sequential transformations and displays have been successfully used along with a whole set of new research techniques developed for the purposes of this study. Over than 250 of different portraits were obtained and depicted eventually for different subsystems belong to the dynamic systems' family under consideration. All portraits were described up to each the so-called invariant cell of the phase portrait, a boundary of it, and features of a phase flow in it, including it's a source and a sink. Results of dynamic systems' behavior research are useful for a wide spectrum of applications such as the mathematical modeling of physical processes, vital problems of civil engineering, for example, in the consideration of seismic stability of buildings etc., computing and producing systems, biological phenomena, and of sociological events. (paper)