Research Article | Open Access
Lyapunov Function and Exponential Trichotomy on Time Scales
We study the relation between Lyapunov function and exponential trichotomy for the linear equation on time scales. Furthermore, as an application of these results, we give the roughness of exponential trichotomy on time scales.
Exponential trichotomy is important for center manifolds theorems and bifurcation theorems. When people analyze the asymptotic behavior of dynamical systems, exponential trichotomy is a powerful tool. The conception of trichotomy was first introduced by Sacker and Sell . They described SS-trichotomy for linear differential systems by linear skew-product flows. Furthermore, Elaydi and Hájek [2, 3] gave the notions of exponential trichotomy for differential systems and for nonlinear differential systems, respectively. These notions are stronger notions than SS-trichotomy. In 1991, Papaschinopoulos  discussed the exponential trichotomy for linear difference equations. And in 1999 Hong and his partners [5, 6] studied the relationship between exponential trichotomy and the ergodic solutions of linear differential and difference equations with ergodic perturbations. Recently, Barreira and Valls [7, 8] gave the conception of nonuniform exponential trichotomy. From their papers, we can see that the exponential trichotomy studied before is just a special case of the nonuniform exponential trichotomy. For more information about exponential trichotomy we refer the reader to papers [9–14].
Many phenomena in nature cannot be entirely described by discrete system, or by continuous system, such as insect population model, the large population in the summer, the number increases, a continuous function can be shown. And in the winter the insects freeze to death or all sleep, their number reduces to zero, until the eggs hatch in the next spring, the number increases again, this process is a jump process. Therefore, this population model is a discontinuous jump function, it cannot be expressed by a single differential equation, or by a single difference equations. Is it possible to use a unified framework to represent the above population model? In 1988, Hilger  first introduced the theory of time scales. From then on, there are numerous works on this area (see [16–23]). Time scales provide a method to unify and generalize theories of continuous and discrete dynamical systems.
In this paper, motivated by , we study the exponential trichotomy on time scales. We firstly introduce -Lyapunov function on time scales. Then we study the relationship between exponential trichotomy and -Lyapunov function on time scales. We obtain that the linear equation admits exponential trichotomy, if it has two -Lyapunov functions with some property; conversely, the linear equation has two -Lyapunov functions, if it admits strict exponential trichotomy. At last, by using these results we investigate the roughness of exponential trichotomy on time scales. Above all, our paper gives a way to unify the analysis of continuous and discrete exponential trichotomy.
This paper is organized as follows. In Section 2, we review some useful notions and basic properties on time scales. Our main results will be stated and proved in Section 3. Finally, in Section 4, we study the roughness of exponential trichotomy on time scales.
2. Preliminaries on Time Scales
In order to make our paper independent, some preliminary definitions and theories on time scales are listed below.
Definition 2.1. Let be a time scale which is an arbitrary nonempty closed subset of the real numbers. The forward jump operator is defined by while the backward jump operator is defined by for every . If , then is called right-dense. And if , then is called left-dense. Let be the graininess function.
For example, the set of real numbers is a time scale with and for and the set of integers is a time scale with and for .
Definition 2.2. A function is called rd-continuous if it is continuous at right-dense points in and left-sided limits exist at left dense points in .
We denote the set of rd-continuous functions by .
Definition 2.3. We say that a function is differentiable at , if for any , there exist -neighborhood of and such that for any one has and is called the derivative of at .
Relate to the differential properties, if and is differentiable at , then we have the following equalities:
Definition 2.4. We say that a function is regressive if for all . Furthermore, if , is called positive regressive. An matrix valued function on time scale is called regressive if is invertible for all .
Let be regressive. Define and for all . Then the regressive set is a Abelian group and it is not hard to verify that the following properties hold:(1); (2); (3); (4), where are regressive.
In order to make our paper intelligible, the exponential function on time scales which we will use in our paper is defined as following. For the more general conception of exponential function on time scales, please refer to .
Definition 2.5. Let be positive regressive. We define the exponential function by for all . Here the integral is always understood in Lebesgue's sense and if , for any one has
From the definition of exponential function we have that if are positive regressive, then(1) and ;(2);(3);(4);(5), where stands for the delta derivative of with respect to .
Lemma 2.6 (L'Hôpital's Rule). Suppose that and are differentiable on . Then if satisfies , and , then the existence of implies that exists and .
3. Exponential Trichotomy and Lyapunov Function on Time Scales
From now on, we always suppose that is a two-sides infinite time scale and the graininess function is bounded, which means that there exists a such that . We consider the following linear equation where is an matrix valued function on time scale , satisfying that is rd-continuous and regressive. To make it simple, in this paper we always require that is regressive. If is not regressive, from  we can know that the linear evolution operator associated to (3.1) is not invertible and exists only for which will make our paper complicated. Here, is called the linear evolution operator if satisfies the following conditions: (1); (2); (3)the mapping is continuous for any fixed .
Equation (3.1) is said to admit an exponential trichotomy on time scale if there exist projections , , for each such that , for and there are some constants and such that for and for We say that (3.1) admits a strong exponential trichotomy on time scale if there exist , , for each and constants , satisfying (3.2), (3.3), (3.4), as well as a constant with and such that Obviously, if for any we have or , then the notion of exponential trichotomy on time scale becomes the exponential trichotomy for differential equation or difference equation, respectively.
Next, we give the notion of Lyapunov function. Consider a function . If the following two conditions are satisfied (H1)for each set and let and be, respectively, the maximal dimensions of linear subspaces inside and , then we have ; (H2)for every , and , we have and ,
then we say that the function is a Lyapunov function.
Let By the notion of Lyapunov function, we can see that there are subspaces and such that . The function is called a -Lyapunov function if is a Lyapunov function and for , satisfies the following conditions: (L1), (L2)for , , (L3)for , , (L4)for , ,
where and is a constant with .
Furthermore, let for each be a symmetric invertible matrix. Set . If is a Lyapunov function (-Lyapunov function), then is called a quadric Lyapunov function (-quadric Lyapunov function). Let . Now, we state and prove our main results.
Theorem 3.1. Suppose that (3.1) has two -quadratic Lyapunov functions, with and with satisfying . If the symmetric invertible matrixes and for and , respectively, satisfy and , then (3.1) has an exponential trichotomy.
In order to prove Theorem 3.1, we need some lemmas.
Lemma 3.2. The subspaces , , , and according to Lyapunov functions and have the following relations
Proof. By the knowledge of mathematical analysis, we can easily get that is a decreasing function for any . Thus, when , we have . Now, we prove this lemma by contradiction. Suppose that there is . Then there are and such that . Since , then by Lemma 2.6 we get
But for , one has
It is a contradiction. So we have .
By the concept of Lyapunov function, we get that for , where or , or and or . Using above inequalities and the similar way by which we get , we can directly obtain that . Here we omit the detailed proofing. Thus, the proof is completed.
When , for , we get and for , we get
When , for , we get and for , we get
Let be projections. Set , and . From Lemma 3.2, we have . Thus, by simply deducing, we get that is also a projection. Set . Next, we will give the boundedness of , , and . By the conditions of Theorem 3.1, we suppose that there is a constant such that and . We have the following lemma.
Proof. Firstly, one has Since then we obtain Similarly, we get
Proof of Theorem 3.1. Firstly, we need to show that there are constants and such that when , one has
and when , one has
For inequality (3.25), by the definition of exponential function, we only need to find a constant such that
Thus, if , (3.25) always holds. Similarly, we get that if , and , then (3.26), (3.27) always hold, respectively. Here, we notice that since is positive regressive, then is significative.
By (3.15)–(3.18) and Lemma 3.3, we get that when , and when , Set , , and . Obviously, we have and the above inequalities imply that for one has and for one has Thus, from the definition of exponential trichotomy on time scales, we get that (3.1) has an exponential trichotomy on time scale . This completes the proof of Theorem 3.1.
Next, we give an approximately converse result of Theorem 3.1.
Proof. Suppose that (3.1) admits a strict exponential trichotomy with projections , , , and constants satisfying (3.2)–(3.5). Set
where and . Let .
When , we get Set . Then if , that is, , we have . By (3.35) we obtain that . Then we get , that is, . That means . Similarly, we have . Thus, is a Lyapunov function. By (3.3), (3.4) and the characters of exponential function on time scales, one has Since , then Thus, Set .
Let and . For , one has Obviously, one has Then by (3.3)–(3.5), we obtain that Thus, Since , then we get Similarly, for , we get
Let . Then when , we get . Let . We get for any , , and when , we get .
Since , then we have and there are constants and with . Thus, when , for any we get that is, and when , for any we get that is, Therefore, is a -Lyapunov function.
Set where and . Let and . Then Let and . Similar to the consideration for , we can see that is also a Lyapunov function satisfying where , .
Since , then we have and there are constants and with . Thus, for any , we obtain that when , that is, and when , that is, Thus, is a -Lyapunov function.
According to all the discussions above we get that when (3.1) has a strict exponential trichotomy, there exists two -Lyapunov functions, with and with satisfying . This completes the proof.
4. Roughness of Exponential Trichotomy on Time Scales
The roughness of exponential dichotomy on time scales had been studied by Zhang and his cooperators in their paper  in 2010. In this section, we go further to study the roughness of exponential trichotomy on time scales, using the results which we get from Section 3. We have known that the notion of exponential trichotomy plays a central role when we study center manifolds, so it is important to understand how exponential trichotomy vary under perturbations. Here, we discuss the following linear perturbed equation where is an matrix valued function on time scale with . For this linear perturbed (4.1) we get the following theorem.
Proof. Since (3.1) has a strict exponential trichotomy, then by Theorem 3.4 there are two -Lyapunov functions, with and with satisfying . Here, and are defined in Theorem 3.4. Let be a solution of (3.1) satisfying . Differentiating on both sides of , one has Let be the linear evolution operator related to (3.1). Then we get . And we can see that is also a solution of (3.1). By (4.2) we get At the same time, by (2.4), (2.7), and (3.46) one has