Several Classes of Plain Dynamic Systems Qualitative Investigation

Abstract

Dynamic systems in applications are useful as mathematical models of those processes and phenomena, where statistical events, or fluctuations, may be disregarded. Dynamic systems may be divided into the two main categories – the systems with continuous time (the flows) and systems with discrete time (the cascades). During the investigations of flows normal autonomous systems of ordinary differential equations are used. The present work is devoted to the original rigorous research of some important family of dynamic systems having reciprocal polynomial right parts, which are the forms of theirs phase variables. The whole wide family under consideration is being split into numeric subfamilies belonging to different hierarchical levels, and is subjected to the first and second Poincare transformations, or mappings. As a result, the full qualitative pattern of trajectories is constructed – using the Poincare sphere – in the Poincare disk. A series of new special investigation methods developed, useful for further investigations of similar dynamic systems’ classes.