On Stability of Discrete Dynamical Systems: From Global Methods to Ergodic Theory Approaches

Abstract

The aim of this paper is to give a complete description of the input–output methods for uniform exponential stability of discrete dynamical systems. We present a new study from four perspectives, in each case providing a deep analysis of the input–output criteria and of the axiomatic structure of the admissible pairs. The first stage is devoted to global conditions of uniform type for stability of discrete variational systems, in the most general case. Next, the results are applied to characterize the stability of discrete nonautonomous systems, without any assumptions on their coefficients. In both cases, the optimality of the methods is motivated by examples, showing that the hypotheses regarding the input and output spaces cannot be removed. The third method is focused on ergodic theory approaches, providing criteria of nonuniform type for stability of discrete variational systems. We characterize the uniform exponential stability by means of nonuniform input–output stabilities relative to ergodic measures, using some very general classes of sequence spaces. Thus, we extend our recent stability results obtained in Dragicevic et al. (J Differ Equ 268:4786–4829, 2020). At the fourth stage, we prove even more, that in certain conditions, in the variational case, the exponential stability can be characterized in terms of stabilities along periodic orbits.