Approaching the Discrete Dynamical Systems by means of Skew-Evolution

The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples.


Introduction
The phenomena of the real world, in domains as economics, biology, or environmental sciences, do not take place continuously, but at certain moments in time.Therefore, a discretetime approach is required.By means of skew-evolution semiflows, we intend to construct a framework that deepens the analysis of discrete dynamical systems.
Playing an outstanding role in the study of stable and instable manifolds and in approaching several types of differential equations and difference equations, the exponential dichotomy for evolution equations is one of the domains of the stability theory with an impressive development.The dichotomy is a conditional stability, due to the fact that the asymptotic properties of the solutions of a given evolution equation depend on the location of the initial condition in a certain subspace of the phase space.Over the last decades, the classic techniques used to characterize asymptotic properties as stability and instability were generalized towards a natural generalization of the classic concept of dichotomy, the notion of trichotomy.The main idea in the study of trichotomy is to obtain, at any moment, a decomposition of the state space in three subspaces: a stable subspace, an instable one, and a third one called the central manifold.We intend to give several conditions in order to describe the behavior related to the third subspace.
A relevant step in the study of evolution equations is due to Henry, who, in [1], studied the property of dichotomy in the discrete setting, in the spirit of the classic theory initiated by Perron in [2].
A special interest is dedicated to the study of dynamic linear systems by means of associated difference equations, as emphasized by Chow and Leiva in [3] and Latushkin and Schnaubelt in [4].
In [5], the uniform exponential dichotomy of discretetime linear systems given by difference equations is presented, and the results are applied at the study of dichotomy of evolution families generated by evolution equations.In [6], a characterization of exponential dichotomy for evolution families associated with linear difference systems in terms of admissibility is given.
In [7], characterizations for the uniform exponential stability of variational difference equations are obtained and, in [8], the uniform exponential dichotomy of semigroups of linear operators in terms of the solvability of discretetime equations over N is characterized.In [9], new characterizations for the exponential dichotomy of evolution families in terms of solvability of associate difference and integral equations are deduced.In [10,11], some dichotomous behaviors for variational difference equations are emphasized, such as a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories or characterizations in terms of the admissibility of pairs of sequence spaces over N with respect to an associated control system.
The notion of trichotomy was introduced in 1976 by Sacker and Sell and studied for the case of linear differential equations in the finite dimensional setting in [12].For the first time, a sufficient condition for the existence of the trichotomy, in fact a continuous invariant decomposition of the state space R  into three subspaces, was given.In the same study, the case of skew-product semiflows was as well approached.
A stronger notion, but still in the finite dimensional case, was introduced by Elaydi and Hajek in [13], the exponential trichotomy for linear and nonlinear differential systems, by means of Lyapunov functions.They prove the fact that the property of exponential trichotomy of a differential system implies the Sacker-Sell type trichotomy.Meanwhile, the notion of invariance to perturbations of the property of trichotomy is given.Thus, in the case of a nonlinear perturbation of a linear exponential trichotomic system, the obtained system preserves the same qualitative behavior as the nonperturbed one.
In [14], a relation between Lyapunov function and exponential trichotomy for the linear equation on time scales is given and, as application, the roughness of exponential trichotomy on time scales is proved.Several asymptotic properties for difference equations were studied in [15][16][17] and, recently, in [18,19].Other asymptotic properties for discrete-time dynamical systems were considered in [20,21].A new concept of   -trichotomy for linear difference systems is given in [22], as an extension of the   -dichotomy and of the exponential trichotomy in   spaces.
The notion of skew-evolution semiflow considered in this paper and introduced by us in [23] generalizes the concepts of semigroups, evolution operators, and skewproduct semiflows and seems to be more appropriate for the study of the asymptotic behavior of the solutions of evolution equations in the nonuniform case, as they depend on three variables.The applicability of the notion has been studied in [24][25][26][27][28].
The case of stability for skew-evolution semiflows is emphasized in [29], and various concepts for trichotomy are studied in [30].Some asymptotic properties, as stability, instability, and trichotomy for difference equations in a uniform as well in a nonuniform setting, were studied by us in [31][32][33].
The following sections outline the structure of this paper.In Section 2, the definitions for evolution semiflows, evolution cocycles, and skew-evolution semiflows are given, featured by examples.In Section 3, we present definitions and characterizations for the properties of exponential growth and decay, respectively, for the exponential stability and instability.The main results are stated in Sections 4 and 5, where we give definitions and characterizations for these asymptotic properties in discrete time for skew-evolution semiflows.Finally, some conclusions are emphasized in Section 6.
The list of references allows us to build the overall context in which the discussed problem is placed.
Definition 1.The mapping  :  ×  →  defined by the relation where  :  ×  →  has the properties The approach of asymptotic properties in discrete time is of an obvious importance because the results obtained in this setting can easily be extended in continuous time.
Example 3. Let  = R be a Banach space and let  = C(N, ) be the set of R-valued sequences (  ) ≥0 .The mapping is an evolution semiflow on .We consider the linear system in discrete time: where  : N → B().If we denote where  : N 2 → B(), then every solution of system (3) satisfies the relation The pair   = (, Φ  ) is a skew-evolution semiflow, associated with system (3), where Φ  is an evolution cocycle over the evolution semiflow , given by where [  ] denotes the integer part of the term of rank .

Preliminary Results
This section aims to emphasize some asymptotic behaviors, as exponential growth and decay and exponential stability and instability, as a foundation for the main results.We give the definitions of these properties in continuous time and we underline the characterizations in discrete time, as results that play the role of equivalent definitions (see [33]).

Proposition 7.
A skew-evolution semiflow  with exponential growth is exponentially stable if and only if there exist a constant  > 0 and a sequence of real numbers (  ) ≥0 with the property   ≥ 1, ∀ ≥ 0, such that for all (, ) ∈ Δ and all (, V) ∈ .
Proof.We have the following.
Necessity.It is obtained immediately if we consider in relation ( 9)  =  and  =  0 = , and if we define where the existence of  : R + → R * + and of ] is given by Definition 6.
Proof.We have the following.
Necessity.We take in relation ( 16)  = ,  = , and  0 =  0 and we define where the existence of function  : R + → R * + and of constant ] is given by Definition 9.
Sufficiency.First step, let us take  ≥  0 + 1 and we denote  = [] and, respectively,  0 = [ 0 ].We obtain for all (, V) ∈ , where the existence of function  and of constant  is assured by Definition 8.As a second step, if we consider  ∈ [ 0 ,  0 + 1), we obtain for all (, V) ∈ .Hence, for all (,  0 , , V) ∈  × , where we have denoted which proves the exponential instability of .

Nonuniform Discrete Dichotomic Behaviors
Definition 11.A projector  on  is called invariant relative to a skew-evolution semiflow  = (, Φ) if the following relation holds for all (, ) ∈  and all  ∈ .
Example 14.We denote by C = C(R + , R + ) the set of all continuous functions  : R + → R + , endowed with the topology of uniform convergence on compact subsets of R + , metrizable relative to the metric is an evolution semiflow on .
In discrete time, we will describe the property of exponential dichotomy as given in the next proposition.

Proposition 15. A skew-evolution semiflow 𝐶 = (𝜑, Φ) is exponentially dichotomic if and only if there exist two projectors
and a sequence of real positive numbers (  ) ≥0 such that for all (, ) ∈ Δ and all (, V) ∈ .
Proof.We have the following.
Proof.We have the following.
Necessity.According to Proposition 7, there exist a constant ] > 0 and a sequence of real numbers (  ) ≥0 with the property   ≥ 1, ∀ ≥ 0. We obtain for  = ]/2 > 0 and according to Proposition 7 where we have denoted By Proposition 10, there exist a constant ] > 0 and a sequence of real numbers (  ) ≥0 with the property   ≥ 1, ∀ ≥ 0. We obtain for  = −]/2 > 0 for all (, ) ∈ Δ and all (, V) ∈ , where we have denoted for all (, V) ∈ , where  and  are given by Definition 5.
We obtain further for  ≥ for all (, V) ∈ , where   = ()    .Then, there exist  ≥ 1 and ] > 0 such that On the other hand, for  ∈ [ 0 ,  0 + 1), we have We obtain that  − is stable, where Hence, there exists a sequence (  ) ≥0 with the property   ≥ 1, ∀ ≥ 0, such that which implies the exponential stability of  1 and ends the proof.
According to the hypothesis, if we consider  =  we obtain for all (, , , V) ∈ Δ × , which implies the exponential instability of  2 and ends the proof.
Example 19.We consider the evolution semiflow  defined in Example 14.Let  = R 3 be endowed with the norm The mapping Φ :  ×  → B(), given by is an evolution cocycle.Then,  = (, Φ) is a skew-evolution semiflow.
In discrete time, the trichotomy of a skew-evolution semiflow can be described as in the next proposition.
Proof.We have the following.
Necessity.( 1 ) is obtained if we consider for  1 in relation (9) of Definition 6  =  and  =  0 =  and if we define (72) Sufficiency.Let  ≥  0 + 1.We denote  = [] and  0 = [ 0 ] and we obtained the relations According to ( 1 ), we have for all (, V) ∈ , where functions  and  are given as in Definition 5.
Hence, the skew-evolution semiflow  is exponentially trichotomic.Some characterizations in discrete time for the exponential trichotomy for skew-evolution semiflows are given in what follows.for all (, ), (, ) ∈ Δ and all (, V) ∈ .
Proof.We have the following.