Investigation of asymptotic stability of equilibria by localization of the invariant compact sets

The method of localization of invariant compact sets was proposed to study for asymptotic stability the equilibrium points of an autonomous system of differential equations. This approach relies on the necessary and sufficient conditions for asymptotic stability formulated in terms of positive invariant sets and invariant compact sets, and enables one to study the equilibrium points for asymptotic stability in the cases where it is impossible to use the first approximation or the method of Lyapunov functions. The possibilities of the method were illustrated by examples.