Notes on the behaviour of trajectories of polynomial dynamic systems

Abstract

Dynamic systems play a key role in various directions of modern science and engineering, such as the mathematical modeling of physical processes, the broad spectrum of complicated and pressing problems of civil engineering, for example, in the analysis of seismic stability of constructions and buildings, in the fundamental studies of computing and producing systems, of biological and sociological events. A researcher uses a dynamic system as a mathematical apparatus to study some phenomena and conditions, under which any statistical events are not important and may be disregarded. The main task of the theory of dynamic systems is to study curves, which differential equations of this system define. During such a research, firstly we need to split a dynamic system’s phase space into trajectories. Secondly, we investigate a limit behavior of trajectories. This research stage is to reveal equilibrium positions and make their classification. Also, here we find and investigate sinks and sources of the system’s phase flow. As a result, we obtain a full set of phase portraits, possible for a taken family of differential dynamic systems, which describe a behavior of some process. Namely polynomial dynamic systems often play a role of practical mathematical models hence their investigation has significant interest. This paper represents the original study of a broad family of differential dynamic systems having reciprocal polynomial right parts, and describes especially developed research methods, useful for a wide spectrum of applications.