Estimation of Regions of Attraction of Dynamical Systems via Polynomial Lyapunov Function

While studying the behavior of a dynamical system, the problem of the stability of a stationary regime plays an important role. This task includes finding the stationary points or limit cycles, determining their stability or instability, and identifying regions of attraction (RoAs) of attractors. There are several classical methods for obtaining RoAs estimates, which can be divided into Lyapunov and non-Lyapunov methods; at the same time, due to the limitations of existing methods, the identification of a complex RoA boundary is practically impossible, and also leads to high computational costs. Existing methods are quite effective for systems of the second and third orders, however, an increase in the dimension of the system or the presence of uncertain mechanical parameters leads to an exponential increase in the required calculations. In this regard, it is important to develop algorithms that are fairly simple in terms of the number of necessary operations and at the same time give acceptable estimates of RoAs from a practical point of view. In this paper, we consider the problem of obtaining estimates for the domains of attraction and stability of a nonlinear dynamical system with a polynomial right hand side. The procedure is illustrated with examples previously studied by various authors.