Abstract and Applied Analysis

Volume 2014 (2014), Article ID 632109, 11 pages

http://dx.doi.org/10.1155/2014/632109

Research Article

Linearization of Impulsive Differential Equations with Ordinary Dichotomy

Yongfei Gao,¹ Xiaoqing Yuan,¹ Yonghui Xia,¹ and P. J. Y. Wong²

¹Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

²School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798

Received 27 November 2013; Accepted 3 January 2014; Published 2 March 2014

Academic Editor: Yongli Song

Copyright © 2014 Yongfei Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper presents a linearization theorem for the impulsive differential equations when the linear system has ordinary dichotomy. We prove that when the linear impulsive system has ordinary dichotomy, the nonlinear system , , , , is topologically conjugated to , , , , where , , represents the jump of the solution at . Finally, two examples are given to show the feasibility of our results.

1. Introduction

A basic linearization theorem is the famous Hartman-Grobman theorem (see [1, 2]). Then Palmer successfully generalized the standard Hartman-Grobman theorem to nonautonomous differential equations (see [3]). Then Fenner and Pinto [4] generalized Hartman-Grobman theorem to impulsive differential equations. Since they did not discuss the Hölder regularity of the topologically equivalent function , for this reason, recently, Xia et al. [5] gave a rigorous proof of the Hölder regularity. Xia et al. [6, 7] proved a version of generalized Hartman-Grobman theorem for dynamic systems on time scales. It should be noted that the abovementioned works are based on the linear differential equations with uniform exponential dichotomy. Therefore, motivated by [8], in this paper, we have a version of generalized Hartman-Grobman theorem for the impulsive differential equations when the linear system has ordinary dichotomy.

Our main objective in this paper is to prove that, when the impulsive linear system has an ordinary dichotomy, the nonlinear system

is topologically conjugated to its linear part

where , , represents the jump of the solution at . Finally, two examples are given to show the feasibility of our results.

2. Definitions

Consider the linear nonautonomous system with impulses at times :

where , , represents the jump of the solution at , , and and are matrixes.

Definition 1. System (3) is said to be an ordinary dichotomy, if there exists a projection and a constant such that

where is a fundamental matrix of linear system (3) and is given by

where is a fundamental matrix of the system , provided that is invertible, for all . In what follows, we will assume that is invertible for all .

Definition 2. In Definition 1, if as , then system (3) is said to possess an ordinary dichotomy with a positive asymptotically stable manifold;

 if as , then system (3) is said to possess an ordinary dichotomy with a negatively asymptotically stable manifold; if both of them hold, then system (3) is said to possess an ordinary dichotomy with asymptotically stable manifolds.

3. Main Result and Proof

Consider the following nonautonomous impulse systems:

where , , represents the jump of the solution at , , and and are matrixes.

Definition 3. Suppose that there exists a function such that

(i)for each fixed , is a homeomorphism of into ;(ii) uniformly bounded with respect to ;(iii)assume that has property (ii) also;(iv)if is a solution of system (7), then is a solution of system (6).

If such a map exists, then system (7) is topologically conjugated to (6). is an equivalent function.

Theorem 4. Suppose that the impulsive linear system (6) has an ordinary dichotomy and for any and one has

(),(),(),(),(),(), ,

where , are integrable functions and , are summable functions in , and and are positive constants. Then system (7) is topologically conjugated to system (6).

Remark 5 (pure continuous case). If impulsive jump operators are absent, then system (6) and (7) reduces to pure continuous systems. That is,

Then Theorem 4 reduces to main results in [8].

Remark 6 (Pure discrete cases). A difference system, or a pure discrete-time system, is a special case of systems with impulses. Thus, instead of (6), we have only

or for the perturbed case (7),

Now, means , so that we may write the linear system in the canonical way as

or similarly for the perturbed system,

where . Then we have the following.

Corollary 7. Suppose that has an ordinary dichotomy, and for any . If satisfies

then system (10) is topologically equivalent to system (9).

Remark 8. We point out that the conditions in Theorem 4 can be approached. For example, taking , if we assume that the interval contains finite number of sequences , then

In particular, if , , then

Before the proof of Theorem 4, let us make some discussions about ordinary dichotomy and introduce some lemmas. Note first that if system (6) has an ordinary dichotomy then a fundamental matrix can be chosen such that the projection in (4). In fact, for any projection , there exists an invertible matrix such that . If (4) holds for and ; then (4) holds for and , this implies that is the required projection if is chosen as a fundamental matrix. Furthermore, in (4), we can assume that with being unbounded on for and bounded on for and with , with . Then is bounded on , is bounded on  (and unbounded on ), and is bounded on  (and unbounded on ).

In what follows, we assume that the assumptions in Theorem 4 always hold. Let be a solution of (7) satisfying the initial condition and a solution of (6) satisfying the initial condition .

Lemma 9. For each , the system

has a unique bounded solution with .

Proof. For each , the solution of system (16) satisfying is

Noting that

On the other hand,

It follows from (17) that

We can assert that and hold. Otherwise, will be unbounded since it concludes unbounded part or/and . Thus

Moreover, if satisfies the initial condition , then

Now, we prove that is bounded. Due to the boundedness of and , we assume that , , where , are some positive constants. Together with (4) and (), it follows that, if , then we have

if , then we have

Similarly, if , then we have

if , then we have

On the other hand, it follows from (4) and () that, if , then we have

If , then we have

Similarly, if , then we have

If , then we have

Therefore, we get

Similarly, we get

It follows from (4), (), (22), and (31) that

So is a bounded solution, and the bounded solution is unique with the initial condition . The proof of Lemma 9 is complete.

Lemma 10. For each , the system

has a unique bounded solution with .

Proof. For a bounded continuous function of whose norm , we define a map as follows:

It follows from (4), (), and (31) that we can also obtain that

So, we have . Therefore, is a self-map of a sphere with radius .

Moreover, it follows from (4), (), (), (), and (32) that

Let be a positive constant such that ; then . By the contraction principle map has a unique fixed point ; that is, satisfies

By direct differentiation, we can verify that is a solution of (34). Furthermore, the solution is bounded with and

Now, we are going to prove that the bounded solution with initial condition (39) is unique. For this purpose, we assume that is another solution of (34). Following steps similar to (17)–(22), it is not hard to show that any bounded solution of (34) with initial value condition (39) can be written as follows:

Calculate , by (4), (), (), (), and (32) that

Hence , and , so we have . This implies that the bounded solution of (34) with initial condition (39) is unique. The proof of Lemma 10 is complete.

Lemma 11. Let be any solution of the system (7); then the system

has a unique bounded solution with .

Proof. Obviously, is a bounded solution of system (42) with the initial condition . Now we show that the bounded solution is unique. If not, there is another bounded solution , by Lemma 10, which can be written as follows:

Then it follows from (4), (), (), (), and (32) that

Since , so with the initial condition . The proof of Lemma 11 is complete.

Let

where is given by (22), and

Lemma 12. Let be any solution of system (7); then is a solution of system (6).

Proof. If is any solution of system (7), then since, by (46), .

We assume that ; then we have

So, is the solution of system (6).

Lemma 13. Let be any solution of system (6); then is a solution of system (7).

Proof. The proof is similar to Lemma 12.

Lemma 14. For any , ,

Proof. According to the above arguments, if is a solution of system (7), from Lemma 12, is a solution of (6). On the other hand, in view of Lemma 13, it is easy to see that is another solution of (7). Let ; we have

Thus is a solution of the system (42). On the other hand, following the definition of , , we can obtain

It follows from Lemmas 9 and 10 that is a bounded solution of the system (42); by Lemma 11, system (42) has only one zero bounded solution with the initial condition . Hence and thus . That is, . Since is arbitrary, we have

Lemma 15. For any , ,

Proof. If is any solution of system (6), from Lemma 13, is a solution of (7). On the other hand, in view of Lemma 12, it is easy to see that is another solution of (6). Let ; we have

Thus is a solution of the system (6). On the other hand, following the definition of , , we can obtain

It follows from Lemmas 9 and 10 that is a bounded solution of the system (6), and it is easy to see that system (6) has only one zero bounded solution with the initial condition ; therefore, and thus . That is, . Since is arbitrary, we have

So and are inverses of each other for each fixed and they are both homeomorphisms for each fixed .

Now we are in a position to prove the main results.

Proof of Theorem 4. We are going to show that satisfies the four conditions of Definition 3.

Proof of Condition (i). For any fixed , it follows from Lemmas 14 and 15 that is homeomorphism and .

Proof of Condition (ii). It follows from and Lemma 9 that is bounded, uniformly with respect to .

Proof of Condition (iii). It follows from and Lemma 10 that is bounded, uniformly with respect to .

Proof of Condition (iv). Following from Lemma 12 and Lemma 13, we easily prove that condition (iv) is true.

Therefore, system (7) is topologically conjugated to system (6). This completes the proof of Theorem 4.

4. Examples

Now we present two examples to show the feasibility of our results. Consider the following impulsive systems:

where

From system (57), we can easily see that satisfies

It is easy to see that the fundamental matrix of is

Then the fundamental matrix of the impulsive linear system (56) is

In what follows, we give two examples of .

Example 1. Taking

denote that is a positive constant such that the interval contains no more than . Then we have

Let

Consequently,

Obviously,

That is, system (56) possesses an ordinary dichotomy. On the other hand, we see that

From Theorem 4, we conclude that (56) and (57) are topologically conjugated.

Example 2. Taking

Taking , we have

That is,

Obviously,

In this case, system (56) also possesses an ordinary dichotomy. On the other hand, we have

From Theorem 4, we conclude that (56) and (57) are topologically conjugated.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant (nos. 11271333 and 11171090) and ZJNSFC.

References

P. Hartman, “On the local linearization of differential equations,” Proceedings of the American Mathematical Society, vol. 14, pp. 568–573, 1963.

D. Grobman, “Homeomorphism of systems of differential equations,” Doklady Akademii Nauk SSSR, vol. 128, pp. 880–881, 1959.

K. J. Palmer, “A generalization of Hartman's linearization theorem,” Journal of Mathematical Analysis and Applications, vol. 41, no. 3, pp. 753–758, 1973.

J. L. Fenner and M. Pinto, “On a Hartman linearization theorem for a class of ODE with impulse effect,” Nonlinear Analysis, Theory, Methods and Applications, vol. 38, no. 3, pp. 307–325, 1999.

Y. H. Xia, X. Chen, and V. Romanovski, “On the linearization theorem of Fenner and Pinto,” Journal of Mathematical Analysis and Applications, vol. 400, no. 2, pp. 439–451, 2013.

Y. Xia, J. Cao, and M. Han, “A new analytical method for the linearization of dynamic equation on measure chains,” Journal of Differential Equations, vol. 235, no. 2, pp. 527–543, 2007.

Y. H. Xia, J. Li, and P. J. Y. Wong, “On the topological classification of dynamic equations on time scales,” Nonlinear Analysis: Real World Applications, vol. 14, no. 6, pp. 2231–2248, 2013.

L. Jiang, “Ordinary dichotomy and global linearization,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 7, pp. 2722–2730, 2009.