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

Abstract


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.

About the authors

Gennady A Rudykh

Institute of Mathematics, Economics and Computer Science of Irkutsk State University

Email: rudykh@icc.ru
(д.ф.-м.н., проф.), профессор, каф. математического анализа и дифференциальных уравнений; Институт математики, экономики и информатики Иркутского государственного университета; Institute of Mathematics, Economics and Computer Science of Irkutsk State University

Daria J Kiselevich

Institute of Mathematics, Economics and Computer Science of Irkutsk State University

Email: dariakis@mail.ru
аспирант, каф. математического анализа и дифференциальных уравнений; Институт математики, экономики и информатики Иркутского государственного университета; Institute of Mathematics, Economics and Computer Science of Irkutsk State University

References

  1. Steeb W.-H. Generalized Liouville equation, entropy, and dynamic systems containing limit cycles // Physica A, 1979. Т. 95, № 1. С. 181-190.
  2. Красносельский М. А. Оператор сдвига по траекториям дифференциальных уравнений. М.: Наука, 1966. 331 с.
  3. Треногин В. А. Функциональный анализ. М.: Физматлит, 2002. 448 с.
  4. Зубов В. И. Динамика управляемых систем. М.: Высш. шк., 1982. 285 с.
  5. Nemytskiy V. V., Stepanov V. V. Qualitative Theory of Differential Equations. Moscow-Leningrad: Gostekhizdat, 1949. 550 p.
  6. Леонов Г. А. Странные аттракторы и классическая теория устойчивости движения. СПб.: СПб. ун-т, 2004. 144 с.

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