Properties of the integral curve and solving of non-autonomous system of ordinary differential equations

• Main Authors: D. J. Kiselevich, G. A. Rudykh
• Format: Article
• Language: English
• Published: Samara State Technical University 2012-06-01
• Series: Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki
• Online Access: http://mi.mathnet.ru/eng/vsgtu1012
• ISSN: 1991-8615; 2310-7081

In this paper, we consider non-autonomous system of ordinary differential equations. For a given non-autonomous system, we introduce the distribution probability-density function of representative points of the ensemble of Gibbs, possessing all the characteristic properties of the probability-density function, and satisfying the partial differential equation of the first order (Liouville equation). It is shown that such distribution probability-density function exists and represents the only solution of the Cauchy problem for the Liouville equation. We consider the properties of the integral curve and the solutions of non-autonomous system of ordinary differential equations. It is shown that under certain assumptions, the motion along trajectories of the system is the maximum of the distribution probability-density function, that is, if all the required terms are satisfied, an integral curve of non-autonomous system of ordinary differential equations at any given time is the most probable trajectory. For the linear non-autonomous system of ordinary differential equations, it is shown that the motion along the trajectories is carried out in the mode of distribution probability-density function and the estimate of its solutions is found.