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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 382592, 11 pages
http://dx.doi.org/10.1155/2013/382592
Research Article

Poincaré Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 12 December 2012; Accepted 1 January 2013

Academic Editor: Yonghui Xia

Copyright © 2013 Yefeng He and Yepeng Xing. 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 is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

1. Introduction

Many systems in physics, chemistry, biology, and information science have impulsive dynamical behavior due to abrupt jumps at certain instants during the evolving processes. This complex dynamical behavior can be modeled by impulsive differential equations. The theory of impulsive differential systems has been developed by numerous mathematicians (see [19]). As to the stability theory and boundary value problems to impulsive differential equations, There have been extensive studies in this area. However, there are very few works on the qualitative theory of impulsive differential equations and impulsive semidynamical systems. Recently, Bonotto and Federson have given a version of the Poincaré-Bendixson Theorem for impulsive semidynamical systems in [10, 11]. As it is known, the method of Poincaré map plays an important role in the research of qualitative theory and is a natural means to study the existence of periodic solutions and its asymptotic stability. However, due to the complexity of the associated impulsive dynamic models, this approach has only been applied successfully to Raibert’s one-legged-hopper (see [1214]) predator-prey models (see [1518]), and so forth. The bifurcation theory for ordinary differential equations or smooth systems appeared during the last decades (see, e.g., [19]); however, little is known about the bifurcation theory of impulsive differential equations due to its complexity (see [20]). In this paper, we mainly study a certain scalar impulsive differential equations on Moebius stripe undergoing impulsive effects at fixed time: where , are fixed with as, , and . Hu and Han (see [20]) investigated the existence of periodic solutions and bifurcations of (1) under the assumptions that and are periodic; that is, the following assumption holds.

(H*) There exist a constant , a positive integer , and two mutual coprime positive integers and such that In this paper, we assume that the following conditions hold.

(H1) Assume that both and are continuous scalar functions on , are odd, continuous functions; that is, .

(H2) There exists a constant , a positive integer such that From (H2), we have is periodic and . Hence assumption (H*) holds naturally. However, we show some new and fruitful results of system (1) with the condition (H1)-(H2). For example, we obtain the existence and stability of periodic solutions to system (1) by double-periodic bifurcation.

This paper is organized as follows. In Section 2, for the sake of self-containedness of the paper, we present some basic definitions of impulsive differential equations. In Section 3, we describe the scalar impulsive differential equations on Moebius stripe and define the Poincaré map. Then we prove several essential lemmas and give sufficient conditions to ensure the existence and stability of one-side and two-side orbits of impulsive differential equation on Moebius stripe. In Section 4, we are mainly concerned with the double-periodic bifurcation impulsive differential equations on Moebius stripe.

2. Preliminaries

For the sake of self-containedness of the paper, we present the basic definitions and notations of the theory of impulsive differential equations we need (see [1, 2, 8]). We also include some fundamental results which are necessary for understanding the theory.

Let , , and be the sets of real numbers, integers, and positive integers, respectively. Denote by a strictly increasing sequence of real numbers such that the set of indexes is an interval in .

Definition 1. A function ,, is from the set if (i)it is left continuous;(ii)it is continuous, except, possibly, points of , where it has discontinuities of the first kind.

The last definition means that if , then the right limit exists and , where , for each .

Definition 2. A function is from the set if , , where the derivative at points of is assumed to be the left derivative.
In what follows, in this section, is an interval in . For simplicity of notation, is not necessary a subset of .

Definition 3. The solution is stable if to any and there corresponds such that for any other solution of (1) with we have for ; the solution is uniformly stable, if can be chosen independently of .

Definition 4. The solution is asymptotically stable if it is stable in the sense of Definition 3 and there exists a positive number such that if is any other solution of (1) with , then as ; if can be chosen to be independent of and is uniformly stable, then is said to be uniformly asymptotically stable.

Definition 5. The solution is unstable if there exist numbers and such that for any there exists a solution , , of (1) such that either it is not continuable to or there exists a moment , such that .
For any , we assume that there exists a , such that ; then the initial value problem (IVP) to first-order impulsive differential equations (1) is given as In what followed, we use to denote the solution of IVP (4).

In [20], Hu and Han investigated system (1) under the assumption (H*) and obtained the following stability results for the periodic solutions.

Theorem 6 (see [20]). Let be a periodic solution of system (1) with period T. If (>1),  then it is uniformly asymptotically stable (unstable), where is the Poincaré map of system (1).

3. Poincaré Map and Periodic Solutions

In this section, we describe the scalar impulsive differential equations on Moebius stripe and define the Poincaré map. Then we prove several essential lemmas and give sufficient conditions to ensure the existence and stability of one-side and two-side orbits (Figure 2) of impulsive differential equation on Moebius stripe.

Lemma 7. Assume that conditions (H1), (H2) hold. Suppose that is a solution of (1) satisfying initial value . Then is also a solution of (1), and

Proof. Let . Then for , we have by (H2) that For , , it follows from (H1), (H2) that , and Thus, we proved that is a solution of (1). On the other hand, it is obvious that Hence, by uniqueness theorem we have that , . This completes the proof.

Let denotes the stripe area on the plain between two lines and ; that is,

Assume that exists for all . Define . In general, we denote () by where , .

It follows from Lemma 7 that has the form

We now introduce an equivalence relation on such that for Then we denote the corresponding quotient space by . From geometric point of view, is obtained by considering two elements and on as the same point (or sticking and together). Thus is a surface with only one side or the well-known Moebius stripe. Obviously, by Lemma 7 the union define a flow on . From this point of view, we call (1) satisfying (H1) and (H2) an impulsive dynamical system on Moebius stripe (see Figure 1).

382592.fig.001
Figure 1: Moebius stripe.
382592.fig.002
Figure 2: Figure of one-side and two-side orbits.

Definition 8 (Poincaré Map). Let be the solution of (IVP) (4). Assume that there exists an interval such that for any , exists on . A map is called a Poincaré map of system (1) if for any

Definition 9. A closed curve is called a one-side periodic orbit on if . And a closed curve is called a two-side periodic orbit on if .

From Definitions 8 and 9, we can easily prove the following assertion.

Lemma 10. One of following alternatives is valid: (i) is a one-side periodic orbit;(ii) is a fixed point of ; that is, ;(iii), .

Proof. We prove it from . Assume (i) is true; that is, is a one-side periodic orbit. Then by Definition 9 we have that that is, . Hence (ii) is valid.
Next, we suppose that (ii) is fulfilled; that is, . Then by Lemma 7 we know
Thus, (iii) is proved.
Finally, if (iii) is true, then .  By the uniqueness of solution of IVP (4), we know
Thus we obtain that is a one-side periodic orbit. The proof is completed.

Similarly, as proof of Lemma 10, we have the following lemma.

Lemma 11. One of following alternatives is valid: (i) is a two-side periodic orbit;(ii) is a 2-periodic point of ; that is, , ;(iii), . And there exists a , such that .

Remark 12. From Lemmas 10 and 11, we see that a one-side periodic orbit must be a two-side periodic orbit since Nevertheless, the converse is not true.
From Remark 12, we give the definition of stability of the mentioned orbits.

Definition 13. Let be a periodic orbit of system (1) (one-side or two-side). Then of system (1) is called stable (asymptotically stable or unstable) if as a periodic solution is stable (asymptotically stable or unstable).

Theorem 14. Assume is the solution of IVP (4) and let . Then is a solution to the following IVP of impulsive differential equations:

Proof. Let and , . Without losing generality, we assume that for some . The solution of IVP can be expressed as Differentiate between both sides of the above equation with respect to , we have Let , then for , is the solution of IVP to ordinary differential equation
Thus
Since is left continuous on , we have
For , is a solution of system where . Thus, we have
Similarly, we have ,
Note We obtain for that Deducing in a similar way, we get where . Then the proof is completed.

By Definitions 9 and (31), we conclude the following assertion.

Corollary 15. Assume that conditions (H1), (H2) hold. Then
As usual, one uses the notion . Then one has

Definition 16. is called a hyperbolic fixed point of if and ; the corresponding one-side periodic orbit is called hyperbolic one-side periodic orbit. If is a two-side periodic orbit with , then we call a hyperbolic two-side periodic orbit.

Theorem 17. Assume that the conditions (H1), (H2) hold. Let be a periodic orbit of system (1) and . Then (i)  implies is asymptotically stable,
(ii) implies is unstable.

Proof. If is a two-side periodic orbit; that is, is a periodic solution of (1). Since both (H1) and (H2) hold, we know that (1) is a periodic impulsive differential equation. Then by (33) and Theorem 6, the conclusion is straightforward.

Example 18. Consider the linear periodic impulsive differential equations on Moebius stripe as follows: where (), , , and there exists a constant , a positive integer ,  such that the following conditions are satisfied:(1) and for ; (2) and are continuous; (3), for all ; (4), for all .

Assume that is a one-side periodic solution of system (34), by the method of variation of constants formula (see [1]), we get

Let ; therefore, we have the following theorem.

Theorem 19. Suppose that (1–4) are satisfied, then(i)there exists a unique one-side periodic orbit for system (34) if , which is asymptotically stable (unstable) provided (),(ii)if , (34) has no two-side periodic orbit. If all the trajectories are two-side periodic orbits expect for a unique one-side periodic orbit.

Proof. For the sake of convenience, we denote
Then
Obviously, has a unique solution for any if , and has a unique solution for any if . Observing that any two-side periodic orbit obtained under the assumption must be a one-side periodic orbit since implies , together with Remark 12, we have (34) has no two-side periodic orbit.
It follows from (36) that . Then by Theorem 6 we have the one-side orbit is asymptotically stable (unstable) provided ().
Next, let . By taking (36) and (37) into account, we have
Suppose that has a unique fixed point , from the above we have , then and . So by taking Lemma 11, is a two-side periodic orbit if .
The proof is ended.

Remark 20. If , in (34); that is, (34) reduces to an ordinary differential equation. We see that . Hence holds automatically, and therefore (34) always has a unique one-side periodic orbit.

Corollary 21. Let ()  be fulfilled and . Then

Now we are in position to consider nonlinear impulsive system on Meobius stripe. To explore the uniqueness of one-side periodic orbit, we induce the following condition.

(H3) Operator , is strictly increasing, for all .

Theorem 22. Suppose that conditions (H1)–(H3) hold, then(i)system (1) has at most one one-side periodic orbit; (ii)if any solution of (1) with is well defined on , then system (1) must has a unique one-side periodic orbit.

Proof. We first prove that system (1) cannot have two one-side periodic orbits. Suppose , and , are two one-side periodic orbits system (1). Then Without losing generality, we assume , then it follows from uniqueness theorem of ordinary differential equations that and cannot intersect when is not an impulsive time. Therefore we have Note is strictly increasing, we get In a similar way, we can prove that , . That is, the curve always stays above curve . This contradicts (42). We put it in another way that Thus, (1) has at most a one-side periodic orbit.
Further, let the solution of system (1) be all defined on . If , the conclusion is proved. We assume that , then we know if . Note We obtain that have opposite signs between  and , and then it follows from the continuity of that there exists such that . Similarly, we can prove has a fixed point in the case of . The proof is completed.

Theorem 23. Assume that conditions (H1)–(H3) hold. Furthermore, suppose there exists a positive number such that Then (1) has a unique one-side periodic orbit.

Proof. From (47) we have that will stay inside for . On the other hand, by (H3), we have that for . Then it follows from (48) that
(see Figure 3).

382592.fig.003
Figure 3: A one-side periodic orbit.

Thus, This implies that is well defined for . By Theorem 22, we obtain that (1) has a unique one-side periodic orbit.

4. Double-Period Bifurcation

In this section, we mainly discuss the bifurcation on periodic orbits. If system (1) has a one-side periodic orbit, without losing generality, we may assume that ; that is, is the one-side periodic orbit. Actually, if is a one-side periodic orbit, then we let ; therefore there exists a transformation of system (1) that

By (H2) and Lemma 10, we know , , , and , for all .

Next, we consider the following perturbed system of system (1): where is with respect to , continuously differentiable with respect to . is with respect to . Moreover, we suppose , , , for all , where . For , we have , . These assumptions mean (H1) and (H2) hold for and , for all . Furthermore, assume that is strictly increasing, then by Theorem 22 we have that system (52) has at most a one-side periodic orbit.

Suppose that (1) has a one-side periodic orbit and . Then by using implicit function theorem in the Poincaré map of system (52), we know that system (52) has a one-side periodic orbit when is sufficiently small. Now let be the solution of system (52) and . Then we can get a transformation of system (52):

By Taylor’s formula, we have where for , ; .

If , then . So and , for , , .

Suppose that is the solution of system (53) with the initial value , is the Poincaré map of system (53). Note Without losing generality, let is a nonhyperbolic solution. That is, and .

Noting that , then by Taylor’s formula, we have where , , and .

Theorem 24. Suppose that and is a one-side periodic orbit of system (52)ε=0 with and . Let . If , then for sufficiently small and  (≤0) implies that system (52) has a unique (no) two-sides periodic orbit near , except for a one-side periodic orbit .

Proof. As before, we obtain that
By our assumption, we have . Therefore, where By the implicit function theorem, there exists a unique function , such that . Therefore, for sufficiently small, there is a unique extremal point near . Moreover, the function takes its minimum (maximum) only if (0): Without loss of generality, we can let and then is the minimum point of . So there exists , such that and for , exists. Therefore, there exists a , such that, for , we have For and , we have From (64), for any , we have And for ,
If , then for all and , we have .
If , then is the unique solution of function .
If , then there exist a unique and a unique , such that Thus system (52) has two (no) two-side periodic orbits if (0). The conclusion is completed (see Figures 4, 5, and 6).

382592.fig.004
Figure 4
382592.fig.005
Figure 5
382592.fig.006
Figure 6

Now we shall calculate and in the simplest case, let . For we can calculate them in the same way. In this case, and , . Suppose () is the solution to system (53) with initial value . For , let

Then for , taking into system (53), we can obtain , , and satisfying the following equations: For , we know

From (69) and (70), we have For , as we know, we get where Clearly, , , and .

Moreover, we know Denote and . Then, Then we can obtain For , we can have . Then where . Therefore,

By considering (76)–(79), we can easily have the following theorem when .

Theorem 25. Suppose that and is a one-side periodic solution of system (52)ε=0 with and . Let If , then for sufficiently small, (0) implies that system (52) has a unique (no) two-side periodic orbit of near , except for a one-side periodic orbit .

By virtue of Theorem 25, we can have the following conclusion.

Corollary 26. (i) Let , (<0). Then is a nonhyperbolic one-side periodic orbit of system (48), which is asymptotically stable (unstable). (ii) Let , , . Then (>0), is a hyperbolic one-side periodic orbit of system (48) , which is asymptotically stable (unstable. Moreover, the two-side periodic orbit is unstable (asymptotically stable) near .

Finally, we give an example to illustrate it.

Example 27. Consider where . It is obvious that for sufficiently small, is strictly increasing, and then is the unique -periodic solution. By direct computation, we have , , , , and . Therefore, , . It follows from Theorem 25 that system (81)  has two (no) -periodic solution of near if is sufficiently small and (0).

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 11271261), Shanghai Natural Science Foundation (no. 12ZR1421600), and Shanghai Educational Committee Project (10YZ74).

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