Definition and Properties of Measures of Stability and Instability of Zero Solution of a Differential System

We consider the problem of the possibility randomly of choosing the initial value for a perturbed solution of a given differential system so that it remained close to the initial zero solution. In this regard, we introduce completely new concepts and study (as applied to different classes of systems) the measure of stability and the measure of instability of various types: Lyapunov, Perron, or upper-limit.