On Stability of Discrete Dynamical Systems: From Global Methods to Ergodic Theory Approaches
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On Stability of Discrete Dynamical Systems: From Global Methods to Ergodic Theory Approaches Davor Dragiˇcevi´c1 · Adina Lumini¸ta Sasu2,3 · Bogdan Sasu2,3
Received: 9 May 2020 / Revised: 8 October 2020 / Accepted: 21 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 Dragiˇcevi´c 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. Keywords Discrete dynamical system · Exponential stability · Input–output system · Ergodic measure · Periodic orbit Mathematics Subject Classification 34D05 · 93D23 · 93D25 · 37C20 · 39A30
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Bogdan Sasu [email protected] Davor Dragiˇcevi´c [email protected] Adina Lumini¸ta Sasu [email protected]
1
Department of Mathematics, University of Rijeka, Radmile Matejˇci´c 2, 51000 Rijeka, Croatia
2
Department of Mathematics, West University of Timi¸soara, V. Pârvan Blvd. 4, Timi¸soara, Romania
3
Academy of Romanian Scientists, Ilfov 3, Bucharest, Romania
123
Journal of Dynamics and Differential Equations
1 Introduction In the past decades, the linear cocycles or skew-product flows played a fundamental role in the asymptotic theory of dynamical systems (see e.g. Chow and Leiva [7–9], Chow and Yi [10], Dragiˇcevi´c [15], Dragiˇcevi´c et al. [16], Huang and Yi [18], Johnson et al. [19], Katok and Hasselblatt [20], Lian and Lu [22], Pliss and Sell [33,34], Sacker and Sell [35–37], Sasu and Sasu [42], Yi [47], Zhou et al. [48]). The notion of cocycle has a long and rich history, that dates back to the famous paper of Oseledets [26]. Substantial contributions to the foundations of the asymptotic theory of linear skew-product flows were brought in the works of Sacker and Sell, where besides the studies regarding the qualitative properties of dynami
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