Stability of systems with random initial data

In this paper we consider a nonlinear non-autonomous system ordinary differential equations (ODE) and the corresponding Liouville equation. Initial data of the ODE system is random and lie in a given region with a known initial distribution law. For non-linear non-autonomous ODE system introduces the concept of $\varepsilon$ is a statistical stability of the solution, which allows us to study the behavior of solutions of the system of ODE's with nondeterministic initial data. Such a study is carried out using the probability density function of distribution of the ensemble of data points in the ODE system. The notion of $\varepsilon$ is a statistical stability of the solution allows to operate directly from the set of trajectories movement of the ODE system, the initial values of which lie in a given area, as well as to test the criterion $\varepsilon$ is a statistical stability rather a function of the probability density distribution of the ensemble of data points in the Gibbs ODE system, which, while satisfying partial differential equation, but it is a linear equation, and moreover sought not the total solution, and the solution of the Cauchy problem. To introduce the notion of $\varepsilon$ is a statistical stability of the solution requires that the nonlinear ODE system has a solution as a whole, ie that the trajectories of the system does not go to infinity in finite time. In the general case, $\varepsilon$ is a statistical stability is not equivalent to the asymptotic Lyapunov stability of solutions. However, between these concepts has close relationship allows us to formulate the necessary and sufficient condition $\varepsilon$ is a statistical stability of the solution for a linear autonomous system of ODE and sufficient condition for the linear non-autonomous system of ODE (for homogeneous and inhomogeneous cases). The study of the dispersion of the nonlinear non-autonomous system of ODE was obtained A necessary and sufficient condition for $\varepsilon$ is a statistical stability of the solution of the ODE system. All the results are illustrated in the examples of content.