Identification of the domain of attraction

In this paper we consider a time-invariant system of differential equations with an equilibrium. We study the problem of a class of Lyapunov functions allowing to describe an estimation of the domain of attraction (DOA). This work is a development of the results obtained by V. I. Zubov. The solution of the problem is based on the application of the second method of Lyapunov. An example illustrates the theoretical results.