Research Article | Open Access
Approaching the Discrete Dynamical Systems by means of Skew-Evolution Semiflows
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.
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 discrete-time 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 , studied the property of dichotomy in the discrete setting, in the spirit of the classic theory initiated by Perron in .
In , the uniform exponential dichotomy of discrete-time 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 , a characterization of exponential dichotomy for evolution families associated with linear difference systems in terms of admissibility is given.
In , characterizations for the uniform exponential stability of variational difference equations are obtained and, in , the uniform exponential dichotomy of semigroups of linear operators in terms of the solvability of discrete-time equations over is characterized. In , 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 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 . For the first time, a sufficient condition for the existence of the trichotomy, in fact a continuous invariant decomposition of the state space 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 , 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 , 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 [15–17] 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 , 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  generalizes the concepts of semigroups, evolution operators, and skew-product 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 [24–28].
The case of stability for skew-evolution semiflows is emphasized in , and various concepts for trichotomy are studied in . 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 [31–33].
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.
2. Preparatory Notions
Let us consider a metric space, a real or complex Banach space, and the family of linear -valued bounded operators defined on . The norm of vectors and operators is . In what follows, we will denote , , and . By we denoted the identity operator on .
Definition 1. The mapping defined by the relation where has the properties, ,, , and satisfies, ,, , , is called skew-evolution semiflow on .
Remark 2. is called evolution semiflow and is an evolution cocycle.
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 be a Banach space and let be the set of -valued sequences . The mapping is an evolution semiflow on . We consider the linear system in discrete time:where . If we denotewhere , then every solution of system (3) satisfies the relationThe pair is a skew-evolution semiflow, associated with system (3), where is an evolution cocycle over the evolution semiflow , given bywhere denotes the integer part of the term of rank .
Example 4. To every skew-evolution semiflow , one can associate the mapping given bysuch that .
3. 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 ).
Definition 5. A skew-evolution semiflow has exponential growth if there exist mappings , , such thatfor all and all .
Definition 6. A skew-evolution semiflow is said to be exponentially stable if there exist a constant and a mapping such that for all and all .
Proposition 7. A skew-evolution semiflow with exponential growth is exponentially stable if and only if there exist a constant and a sequence of real numbers with the property , , such thatfor all and all .
Proof. We have the following.
Necessity. It is obtained immediately if we consider in relation (9) and , and if we definewhere the existence of and of is given by Definition 6.
Sufficiency. As a first step, if , we denote and . The following relations hold:We obtainfor all , where functions and are given by Definition 5.
As a second step, for , we havefor all .
Hence, is exponentially stable.
Definition 8. A skew-evolution semiflow has exponential decay if there exist mappings , , such thatfor all and all .
Definition 9. A skew-evolution semiflow is said to be exponentially instable if there exist a mapping and a constant such thatfor all and all .
Proposition 10. A skew-evolution semiflow with exponential decay is exponentially instable if and only if there exist a constant and a sequence of real numbers with the property , , such thatfor all and all .
Proof. We have the following.
Necessity. We take in relation (16) , , and and we definewhere the existence of function and of constant is given by Definition 9.
Sufficiency. First step, let us take and we denote and, respectively, . We obtainIt follows thatfor all , where the existence of function and of constant is assured by Definition 8.
As a second step, if we consider , we obtainfor all .
Hence,for all , where we have denotedwhich proves the exponential instability of .
4. Nonuniform Discrete Dichotomic Behaviors
Definition 11. A projector on is called invariant relative to a skew-evolution semiflow if the following relationholds for all and all .
Definition 12. Two projectors and are said to be compatible with a skew-evolution semiflow ifprojectors and are invariant on ;for all , the projections and verify the relations
Definition 13. A skew-evolution semiflow is called exponentially dichotomic if there exist functions , , constants , , and two projectors and compatible with such thatfor all and all .
Example 14. We denote by the set of all continuous functions , endowed with the topology of uniform convergence on compact subsets of , metrizable relative to the metric If , then for all we denote , . Let be the closure in of the set , where is a nondecreasing function with the property . Then, is a metric space and the mapping is an evolution semiflow on .
Let be endowed with the norm The mapping , given by is an evolution cocycle over the evolution semiflow . We consider the projectorsAswe obtain thatSimilarly, asit follows thatThe skew-evolution semiflow is exponentially dichotomic with characteristics
In what follows, let us denote , where and
In discrete time, we will describe the property of exponential dichotomy as given in the next proposition.
Proposition 15. A skew-evolution semiflow is exponentially dichotomic if and only if there exist two projectors , compatible with , constants , and a sequence of real positive numbers such that for all and all .
Proof. We have the following.
Necessity. If we consider for in relation (26) of Definition 13 and and if we definerelation is obtained.
Statement results from Definition 13 for if we consider in relation (27) and andSufficiency. It is obtained by means of Proposition 7 for and, respectively, of Proposition 10 for .
Hence, is exponentially dichotomic.
Theorem 16. A skew-evolution semiflow is exponentially dichotomic if and only if there exist two projectors , compatible with such thatthere exist a constant and a sequence of real positive numbers such thatthere exist a constant and a sequence of real positive numbers such that for all and all .
Proof. We have the following.
Necessity. According to Proposition 7, there exist a constant and a sequence of real numbers with the property , . We obtain for and according to Proposition 7where we have denotedBy Proposition 10, there exist a constant and a sequence of real numbers with the property , . We obtain for for all and all , where we have denotedSufficiency. Let , . We define and . We consider , where we define . We have thatfor all , where and are given by Definition 5. We obtain further for for all , where . Then, there exist and such thatOn the other hand, for , we haveWe obtain that is stable, whereHence, there exists a sequence with the property , , such thatwhich implies the exponential stability of and ends the proof.
According to the hypothesis, if we consider we obtainfor all , which implies the exponential instability of and ends the proof.
5. Nonuniform Discrete Trichotomic Behaviors
Definition 17. Three projectors are said to be compatible with a skew-evolution semiflow ifeach projector , is invariant on ;for all , the projections , , and verify the relations
Definition 18. A skew-evolution semiflow is called exponentially trichotomic if there exist the functions , , , , constants , , , and with the propertiesand three projectors , , and compatible with such thatfor all and all .
Example 19. We consider the evolution semiflow defined in Example 14. Let be endowed with the norm The mapping , given by is an evolution cocycle. Then, is a skew-evolution semiflow.
We consider the projections The skew-evolution semiflow is exponentially trichotomic with characteristics
As in the case of dichotomy, let , where and
In discrete time, the trichotomy of a skew-evolution semiflow can be described as in the next proposition.
Proposition 20. A skew-evolution semiflow is exponentially trichotomic if and only if there exist three projectors compatible with , constants , , , and with the property , and a sequence of positive real numbers such that for all and all .
Proof. We have the following.
Necessity. is obtained if we consider for in relation (9) of Definition 6 and and if we define follows according to Definition 9 for if we consider in relation (16) , , and and is obtained for out of relation (8) of Definition 5 for , , and and if we define follows for from relation (15) of Definition 8 for , , and and if we considerSufficiency. Let . We denote and and we obtained the relationsAccording to , we havefor all , where functions and are given as in Definition 5.
For , we havefor all . Hence, relation (57) is obtained.
Let and and, respectively, . It follows thatFrom , it is obtained thatfor all , where and are given by Definition 8.
For , we havefor all . It follows thatfor all , where we have denotedand which implies relation (58).
By a similar reasoning, from relation (59) is obtained, and from relation (60) follows.
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.
Theorem 21. A skew-evolution semiflow is exponentially trichotomic if and only if there exist three projectors compatible with such that has exponential growth, and has exponential decay and such that the following relations hold:there exist a constant and a sequence of positive real numbers such thatthere exist a constant and a sequence of positive real numbers such thatthere exist a constant and a sequence of positive real numbers such thatthere exist a constant and a sequence of positive real numbers such that for all and all .
Proof. We have the following.
Necessity. As is exponentially trichotomic, Proposition 20 assures the existence of a projector , of a constant , and of a sequence of positive real numbers such holds. Let . If we considerwe obtain where we have denotedSufficiency. As has exponential growth, there exist constants and such that relationholds for all and all . If we denote , the inequality can be written as follows:We consider successively the relations. By denoting , we obtain for all and all . If we define the constantand the sequence of nonnegative real numbersrelation is obtained.
Similarly, the other equivalences can also be proved.
In order to characterize the exponential trichotomy by means of four projectors, we give the next definition.
Definition 22. Four invariant projectors that satisfy for all the following relations, and ,,,,are called compatible with the skew-evolution semiflow .
Theorem 23. A skew-evolution semiflow is exponentially trichotomic if and only if there exist four projectors compatible with , constants , and a sequence of positive real numbers such that