Investigation of the stability of trajectories for three-dimensional nonlinear dynamic system using an accompanying coordinate reference

Abstract

The development of methods of stability analysis and the study of various types of dynamic systems stability belongs to an urgent scientific direction. Among the problems solved in the framework of this direction, an important place is occupied by the problems of finding conditions for stability and stabilization in the sense of N.E. Zhukovsky of trajectories of dynamic systems modeled by systems of three nonlinear differential equations of the first order. The aim of the work is to study the stability in the Zhukovsky sense of the trajectories of a dynamic system described by three autonomous nonlinear first-order differential equations using a differential geometric method called the accompanying coordinate reference method. The formulation of the stability problem in the Zhukovsky sense of the trajectories of dynamic systems described by systems of three nonlinear differential equations of the first order is considered. The definitions of stability of trajectories of a three-dimensional nonlinear system are specified. Using the accompanying coordinate reference, the conditions of exponential stability and instability according to Zhukovsky are obtained. The results can be used in studying the stability of processes in systems of natural science and technology, as well as in solving theoretical and applied problems of mathematical modeling and qualitative research of trajectories of nonlinear dynamic systems.