#### Abstract

Motivated by the importance and application of discrete dynamical systems, this paper presents a new Lyapunov characterization which is an extension of conventional Lyapunov characterization for multistable discrete-time nonlinear systems. Based on a new type stability notion of -stability introduced by D. Efimov, the estimates of solution and the Lyapunov stability theorem and converse theorem are proposed for multi-stable discrete-time nonlinear systems.

#### 1. Introduction

Theory of discrete-time systems is rapidly developed and widely applied to various fields (see three remarkable books [1–3]). In paper [4], motivated by a continuous second-order predator-prey ecological system of Lotka-Volterra type, Efimov introduces a new type notion, -stable and presents Lyapunov characterization for multistable continuous nonlinear systems. For ecological systems, they usually considered that evolution and translation of populations is continuous. Thus continuous models [5, 6] are considered in many references. However, according to observing the translation process of population change, discrete models are better to represent ecological systems [7, 8]. Thus it is meaningful to study -stability for discrete dynamical systems. This paper extends stability results given by Efimov about continuous multistable systems to discrete-time multistable systems.

Stability analysis is one of the main issues for research of control systems theory. A rapid progress has been made in local or global stability analysis for unique equilibrium [9], trajectories [10], close invariant set [11, 12], part of state variable [13], and so forth. In papers [14, 15], Sontag and Wang introduce Lyapunov characterization of input to state stability for continuous systems. Paper [16] by Jiang and Wang presents the property of input to state stability for discrete-time systems. Converse Lyapunov theorem is presented in paper [11] by Lin et al. and paper [17] by Jiang and Wang for continuous and discrete-time systems, respectively. In recent years, multistable systems have attracted considerable attention [8, 18–21]. There have many methods to deal with stability problem for multistable systems. Two popular modern approaches are based on density functions [19] and monotone systems [20]. The former approach substitutes conventional Lyapunov function with density function for establishing stability of stable set. The latter approach develops some constructive conditions based on monotone systems for establishing stability of the set of equilibriums. The above approaches are effective to handle the stability problem of multistable systems. The stability results obtained according to the above approaches are based on conventional stable notions. However, in the areas of theoretical biology and engineering, many systems that represent models are called multistable systems. The set of all invariant solutions of those systems contains stable subset and unstable subset.

In this paper, we firstly introduce the notion of -stability to discrete-time multistable systems. Using some important approaches and techniques of stability analysis in papers [4, 11, 14–18], new Lyapunov characterizations are proposed for discrete-time multistable systems. Based on notions of -Lyapunov function and weak -Lyapunov function, the relation of two functions is presented, and a converse Lyapunov theorem is proved. Our main contribution is that Lyapunov characterizations presented in this paper contain the conventional Lyapunov characterization and it should be extensively applied.

The rest of the paper is organized as follows. Problem statement and mathematical preliminary are presented in Section 2. Stability results of multistable discrete-time nonlinear systems are proposed in Section 3. Finally, a brief conclusion is provided to summarize the paper in the final section.

#### 2. Problem Statement and Mathematical Preliminary

Consider the following discrete nonlinear system: where is the system state vector at time instant , , is locally Lipschitz continuous. denotes the solution for any initial value , and . denotes the Euclidean norm of vector .

Let be a nonempty subset of . The set is called (forward) invariant for system (2.1) if An invariant set is called minimal if it does not contain other smaller invariant sets. The distance of the set from a point is defined as

An invariant set is said to be a locally attracting set if there exists an open neighborhood of such that, for any , . An invariant set is said to be a locally repelling set if exists an open neighborhood of such that, for any , .

Let be the set of all invariant solutions of system (2.1). Clearly, it is an invariant set. Assume the set is a closed and compact set and satisfies , where and denote attracting set and repelling set, respectively.

We first introduce the notion of -asymptotical stability [4] for continuous nonlinear systems to discrete-time nonlinear systems (2.1).

*Definition 2.1. *The system (2.1) is called -stable with respect to if, for some given constant and for each , there exists , such that when , it holds

*Remark 2.2. *It is possible to exist unstable equilibriums in the set which contains all invariant solutions. When the trajectories of system initiate from a neighborhood of unstable equilibriums, it cannot ensure the trajectories in this neighborhood. Thus, the definition of -stability need the existence such that . If is not empty then . Particularly, when is empty (it implies ), -stability is reduced to conventional stability (see [1, 2, 9]). The constant can be related with the radius of the set [4].

*Definition 2.3. *The system (2.1) is called -asymptotically stable with respect to if(i)it is -stable;(ii)it satisfies -attracting property. There is a positive constant , such that, for all , ; that is, for each and , there exists such that

*Remark 2.4. * is dependent on , and . is different for different initial values; that is, there exists no uniform time of convergence to of the trajectories which start from the neighborhood of different initial values due to the presence of unstable equilibriums.

*Example 2.5. *Consider a second-order discrete system
The set of all invariant solutions is . Let . Simulation results are shown in Figure 1.

**(a)**

**(b)**

According to Figure 1, we can get that is unstable and and is asymptotically stable. Thus , and . System (2.6) is -asymptotically stable by Definition 2.3.

*Example 2.6. *Consider a second-order predator-prey system with a prey refuge in the following form:
where , represent the prey and predator density, parameters and are the intrinsic growth rates of the prey and the predator, respectively. is the step size. is a refuge protecting coefficient of the prey. The rest coefficients are positive constants. Let , , and . The set of all invariant solutions of system (2.7) is . Simulation results are shown in Figure 2.

**(a)**

**(b)**

According to Figure 2, the trajectories of solutions from the neighborhood of points (0,0) and (2,0) are convergent to point . We have and are unstable and is asymptotically stable. System (2.7) is -asymptotical stable.

*Remark 2.7. *For system (2.7), there have been many important and interesting results, such as the global stability, periodic solutions, almost periodic solutions, and chaos (see [4–8, 14, 15, 22]). Here our interesting focus is on -asymptotical stability of the set of all invariant solutions.

Throughout this paper, assume and are not empty. In this case, we can exclude an open set containing from admissible set of initial value. Then there exists a uniform convergent time in the definition of attracting property. That is, -attracting property is similar to the definition of attracting property in [9, 18]. Define a hyper-surface Then we obtain Let an open neighborhood of be . We can choose some which ensure the properties , hold only for .

*Definition 2.8. *System (2.1) satisfies -attracting property. Choose any open set of . For each and there exists such that for any

#### 3. Stability Results

##### 3.1. Estimates of Solution

Lemma 3.1. *The system (2.1) is -stable with respect to for some given if and only if for any constant there exists a class function such that for any **
where .*

*Proof. **Sufficiency.* For system (2.1), when , (3.1) holds for any . For each , choose , when , holds for any . -stability is ensured.*Necessity.* Assume system (2.1) is -stable, that is, for some given and for each , there exists , such that
For fixed and , let be the supremun of all applicable . Clearly, the function is positive and nondecreasing. So there exist a class function and a constant such that , where . Let . Then is a function of class . Let . Given , let . Then we have and . Thus -stability implies the property as in inequality (3.1).

*Remark 3.2. *The construction approach of class function is similar to its in [9, 11]. When , the result of Lemma 3.1 is the same as the corresponding result in [9, 11].

Lemma 3.3. *The system (2.1) is -asymptotically stable with respect to if and only if for some constant and any constant there exist a class function and a class function such that for any *

*Proof. **Sufficiency.* Suppose there is a class function such that inequality (3.3) is satisfied. With fixed , we have
By Lemma 3.1, system (2.1) is -stable. For any , it yields
It implies as . Attracting property is satisfied. Thus, system (2.1) is -asymptotically stable.*Necessity.* Suppose that system (2.1) is -asymptotically stable. According to Lemma 3.1, there exists a class function such that for any
Moreover, choose an arbitrary small constant which satisfies . For any and given , there exists such that
Let be the infimun of . The function is nonnegative in and nonincreasing in , and for all . Let
The function is positive and has the following properties:for each fixed , is continuous, strictly decreasing, and as ;for each fixed , is strictly increasing in .Take . Then also satisfies the above two properties and . So
According to inequalities (3.6) and (3.9) we get
Thus, according to arbitrariness of , there exists a class function such that inequality (3.3) is satisfied.

##### 3.2. Stability Theorem

*Definition 3.4. *A continuous function is a -Lyapunov function with respect to for system (2.1) ifthere exist class functions and and a constant such that for any and there exists a class function such that for any and Assume system (2.1) has output , where is a continuous function. The system (2.1) with output is -detectable if for any and

*Definition 3.5. *A continuous function is a weak -Lyapunov function with respect to for system (2.1) ifthere exist class functions and and a constant such that for any and there exists a continuous function , with and for all such that for any and system (2.1) with output is -detectable.

Theorem 3.6. *Weak -Lyapunov function implies -asymptotical stability.*

*Proof. *Suppose system (2.1) has a weak -Lyapunov function. Using inequality (3.15), we have
It implies is bounded and
Thus
According to Lemma 3.1, system (2.1) is -stable with respect to .

Since the compactness of set , there exist a class function and a positive constant such that
which shows is bounded. Thus, for any solution of system (2.1) we can find a forward invariant attracting compact set . Choosing , we have for any .

Furthermore, by we get is nonincreasing. However, by we obtain is nondecreasing. Thus, there exists a positive constant such that
which implies
Then . Using the detectability of system (2.1) with , we have as . That is, -attracting property holds.

The proof is completed.

Lemma 3.7 is given by Sontag in [15] which is useful for proof of Theorem 3.8.

Lemma 3.7. *Assume that is a function of class . Then there exist two class functions and such that
*

Theorem 3.8. *Considering system (2.1), the following is equivalent: ** there exists a -Lyapunov function;**there exists a weak -Lyapunov function;**there is -asymptotically stable.*

*Proof. *(a)*⇒*(b). Let . Because is a class function, is continuous and inequality (3.15) is satisfied. Furthermore, when , we have
The Property (iii) of Definition 3.5 holds. Thus (a) implies (b).

(b)*⇒*(c). The proof is given in Theorem 3.6.

(c)*⇒*(a). Assume that system (2.1) is -asymptotically stable. According to Lemma 3.3, there exist a class function and a class function such that for any
By Lemma 3.1, there exist two class functions and such that
Let . We have
Define . Clearly, is continuous since is a class function. By the definition of , it yields
satisfies the property of inequality (3.11) due to inequality (3.27).

In the following we show satisfies the property as in inequality (3.12). Arbitrarily choose and . Then and
Since function is Lipschitz continuous, the solution of system (2.1) is unique for arbitrary initial value. Then we have
Considering (3.28), we can get
That is,
Due to the arbitrariness of , we have
Function satisfies the property as in inequality (3.12).

#### 4. Conclusion

We conclude with a brief discussion. The notion of -stability introduced by Efimov is different from conventional notion of stability. It is required to consider the set of all invariant solutions of systems. However, the set of all invariant solutions can contain not only stable invariant solutions but also unstable invariant solutions. If it does not contain unstable invariant solutions, -stability is conventional stability. Thus our results should have an more extensive application than those corresponding results in the sense of conventional stability.

#### Acknowledgment

This work was supported by the Natural Science Foundation of China under the Contract no. 60874006.