New criteria of equilibrium points stability for discrete-time systems

An investigation of the equilibrium points stability of dynamical systems is based on the analysis of the linear approximation or on the construction of a Lyapunov function. However, the first method is not applicable in the degenerate case, and the second one requires a construction of functions of a special kind, which is not always possible. At the same time, the Lyapunov function method is based on the notion of an invariant set. Therefore, an idea arises to analyze the stability of the equilibrium point by direct verification of the invariance of certain sets in phase space. In this paper, we are discussing criteria for stability and asymptotic stability of equilibrium points of autonomous discrete-time systems in terms of invariant sets. We also propose methods for verifying these criteria.