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Computational and Mathematical Methods in Medicine
Volume 2013 (2013), Article ID 582820, 9 pages
http://dx.doi.org/10.1155/2013/582820
Research Article

Bifurcations and Stability of Nondegenerated Homoclinic Loops for Higher Dimensional Systems

1Science College, Linyi University, Linyi, Shandong 276005, China
2School of Mathematics Science, Shandong Normal University, Jinan 250014, China

Received 10 August 2013; Revised 23 September 2013; Accepted 23 September 2013

Academic Editor: Jinde Cao

Copyright © 2013 Yinlai Jin 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

By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence of -homoclinic loop and -periodic orbit is obtained. Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.

1. Introduction

With the rapid development of nonlinear science, in the studies of many fields of research and application of medicine, life sciences and many other disciplines, there are a lot of variety high-dimensional nonlinear dynamical systems with complex dynamic behaviors. Homoclinic and heteroclinic orbits and the corresponding bifurcation phenomenons are the most important sources of complex dynamic behaviors, which occupy a very important position in the research of high-dimensional nonlinear systems. We know that in the study of high-dimensional dynamical systems of infectious diseases and population ecology we tend to ignore the stability switches and chaos when considered much more the nonlinear incidence rate, population momentum, strong nonlinear incidence rate, and so forth. The existence of transversal homoclinic orbits implies that chaos phenomenon occur; therefore, it is of very important significance to study the cross-sectional of homoclinic orbits and the preservation of homoclinic orbits for the system in small perturbation.

In addition, in the study of infectious diseases and population ecology systems, we sometimes require the existence of periodic orbits. And, homoclinic and heteroclinic orbits bifurcate to periodic orbits in a small perturbation means that we can get the required periodic solution only by adding a small perturbation when using the similar system which exists homoclinic or heteroclinic orbits to represent the natural system. This also explains the importance of homoclinic and heteroclinic orbits bifurcating periodic orbits in real-world applications.

Therefore, by using the research methods and theoretical results of qualitative and bifurcation problems of high-dimensional systems, especially the results of homoclinic and heteroclinic orbits and their bifurcations for the systems, to study the high-dimensional infectious disease dynamics and population ecology systems to reveal the complex dynamical behavior of the nonlinear dynamical systems and the corresponding reality systems is essential.

About the study of bifurcation problems of homoclinic and heteroclinic loops for two-dimensional systems, a large number of papers were obtained and achieved many good results (for some results see [16]); but for higher-dimensional nonlinear systems, due to the complexity, the results we see today are not abundant. Chow S. N., Deng B., and Fiedler B. discussed the bifurcation of non-degenerated homoclinic loop [7], but the research method is abstract, and the results are focused on the theory. Some subsequent studies are mostly based on the traditional Poincaré map construction method. Zhu [8] discussed the non-degenerated bifurcation problems of homoclinic loop of the system . Compared with [7], paper [8] described the bifurcation surface and bifurcation phenomenon by using the inherent eigenvalue, so that the results possess good operability.

In this paper, the bifurcation and stability problems of non-degenerated homoclinic loops under non-resonant and resonant conditions for high dimensional system are considered. The method to establish the local coordinates system in the tubular neighborhood of homoclinic loop used in [8] is simplified here. In [8], the author used the generalized Floquet method to establish local coordinate systems and Poincaré map. Here, we use the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local coordinates system of the perturbed system in the small neighborhood of the homoclinic orbit. The Poincaré Maps and bifurcation equations obtained by this method are more simple and convenient for analysis than [8]. Besides, this method does not only have important significance in theory, but it can also be operated well in applications.

2. Hypotheses

Suppose the following system: where , satisfies the following hypotheses.(H1) (Hyperbolicity) is a hyperbolic equilibrium of system (1), the stable manifold and the instable manifold of are -dimensional and -dimensional, respectively. Moreover, has simple eigenvalues , , such that, any remaining eigenvalue of satisfies either or , for some positive numbers and .(H2) (Nondegeneration) System (1) has a homoclinic loop , for any , codim .(H3) (Strong inclination) where, and are the strong stable manifold and the strong instable manifold of , respectively, is the generalized eigenspace corresponding to those eigenvalues with smaller real part than , and is the generalized eigenspace corresponding to those eigenvalues with larger real part than . Let , and be unit eigenvectors corresponding to and , respectively. and .

Now, we consider the bifurcation problems of the following perturbation system: where , , , and .

3. Local Coordinates

Suppose that the neighborhood of is small enough and (H1)(H3) are established, then, for is small enough, we can introduce a change such that system (3) has the following form in : where , , , , , , , , , and means transposition. Moreover, in , we suppose that the instable manifold, the stable manifold, the strong instable manifold, the strong stable manifold and the local homoclinic orbits have the following forms, respectively. where, , , , .

Denote . Taking a time transformation if necessary, we may assume , , , where is small enough such that , , , .

Consider the linear system

Similar to [911], system (6) has a fundamental solution matrix , satisfying where , , , , and for small enough, , ; , ; , ; , .

Thus, we may select as a local coordinate system along .

Denote , . So, is a fundamental solution matrix of the adjoint system of (6). And, is bounded and tends to zero exponentially as [8, 9, 12, 13].

Denote ,

We say that is nontwisted as , and twisted as . In this paper, we only consider the nontwisted case.

4. Poincaré Maps and Bifurcation Equations

Denote ,  ,  , . Let , be the cross sections of at and , respectively, where, is small enough such that , Figure 1.

582820.fig.001
Figure 1

Now, we set up Poincaré map , where, the map is definited by the approximate solution of (3) in the small neighborhood of (similarly, we can set up the map by the flow of the linear system of (4) in the small neighborhood of ), is set up by the solution manifold of (3) in the tubular neighborhood of .

4.1. The Relationship between the Two Kinds of Coordinates

Denote , , , , and . Now, we can set up the relationship between the two kinds coordinates and of , . Let where , , .

Using the expressions of , , and by simple calculation, we have , , and where .

4.2. Set Up

Substitute into (3), and use , , we have

Multiplying both sides by , and using , we have , that is,

Thus, from the flow of (14), we define the map , which has the following form where , , is said to the Melnikov vectors [8, 9, 14, 15].

4.3. Set Up

Now, we consider the map , defined by the orbit of (4). For convenience, we assume that . Obviously, for small enough, .

Let be the flying time from to , and be called Silnikov time. By (4), in , we get

Neglecting the higher order terms, the Poincaré map : from to defined by (16) as follows:

is called Silnikov coordinate.

4.4. and Bifurcation Equation

Let , by (15) we have

Substituting (11), (12), and (17) into (18), and neglecting the higher order terms, the Poincaré map is given by

So, from the above and (11), we have the successor function as follows:

Thus, for , there is a one to one correspondence between the 1-homoclinic loop and 1-periodic orbit of (3) and the solution of the following equation:

Equation (21) is called bifurcation equation.

5. Nonresonant Bifurcations

(H4) (Nonresonant condition) .

Obviously, for small enough, we may assume .

Theorem 1. Suppose that hypotheses (H1)~(H4) are valid, then if is small enough, the system (3) exists no more than one 1-homoclinic loop or one 1-periodic orbit in the neighbourhood of . Moreover, the 1-homoclinic loop and the 1-periodic orbit cannot coexist.

Proof. Consider the solution of (21). Let . From (20), we have , where elements of are all zero except . Therefore, . According to the implicit function theorem, in the neighbourhood of , (21) exists a unique solution which satisfies , , .
If , then the solution (22) corresponds to a 1-homoclinic loop of the system (3), that is, the homoclinic loop is persistent.
If , then the solution (22) corresponds to a 1-periodic orbit of the system (3), that is, the homoclinic orbit bifurcates to a periodic orbit.
The proof is complete.

Theorem 2. If , then there exists a -dimensional surface in the small neighbourhood of , such that when and , the system (3) exists a unique homoclinic loop near . If , then the system (3) exists a unique periodic orbit near . If , then the system (3) has no any homoclinic loop and periodic orbit near . is called bifurcation surface, its analytical expression is .

Proof. From (20), for and small enough, equations and always have a unique solution , . Substituting into , we get
If , then, according to the implicit function theorem, in the neighbourhood of , the equation defines a unique -dimensional surface , such that if and , (23) has the solution , the uniqueness can be obtained by the Theorem 1.
If , then (23) has the small positive solution .
If , then (23) has nonzero negative solution . At present, the system (3) has neither homoclinic loop no periodic orbit near the neighbourhood of .
The proof is complete.

Now, we consider the nonexistence of -homoclinic loop and -periodic orbit, where . Firstly, we consider the case of .

We rewrite the time from and as , . Suppose , , and let be the time from to , .

Similar to the previous discussion, we can get its associated successor function as follows:

Denote , , then , where the elements of are all zero except , .

Hence, . According to the implicit function theorem, near , the bifurcation equation has a unique solution which satisfies , , , , and , .

If , then the homoclinic loop of the system (3) which the solution (26) corresponds to is the 1-homoclinic loop.

Because 1-periodic orbit obviously,corresponds to the solution of (26) then, by the uniqueness of solution, the system (3) has no 2-periodic orbit.

If , , or , , then and will get the contradiction.

And, so, for any , we have the following.

Theorem 3. Suppose that (H1)(H4) are fulfilled, , then the system (3) does not have any -homoclinic loop and -periodic orbit for sufficiently small.

6. Resonance Bifurcation

We say that the homoclinic loop is Resonance if . For convenience, we assume the resonant condition has the following form.(H5) (Resonant condition) , , , where , , and .

At first, we discuss the bifurcations of 1-homoclinic loop and 1-periodic orbit. Now, the bifurcation equation has the following form: Similarly for , there is a one to one correspondence between the 1-homoclinic loop and 1-periodic orbit of (3) and the solution of the bifurcation equation (27). It is Easy to see that, for the sufficiently small and , equations , of (27) always have a unique solution , . Substituting it into , we get

Denote , , we have the following conclusion.

Lemma 4. Suppose (H1)(H3) and (H5) are fulfilled, then, for , the necessary condition of and are tangent at is that . Meanwhile, if and , , and are tangent at if and only if , and

Proof. and are tangent at if and only if , , that is,
Solving the above equations, we have . Substituting it into (30), (29) is fulfilled.
The proof is complete.

Suppose is the surface defined by (29), is the surface defined by (28), . Besides, if , (28) is turned to .

Theorem 5. Suppose (H1)(H3) and (H5) are fulfilled, , . If , then, in the neighborhood of , there exists two -dimensional surfaces and , such that, for sufficiently small , we have the following.(i) The system (3) has a unique 2-fold 1-periodic orbit near if and only if .(ii) The system (3) has no -homoclinic orbit and -periodic orbit near if and only if satisfies .(iii) The system (3) has exactly two 1-periodic orbits near if and only if satisfies .(iv) The system (3) has exactly a -homoclinic orbit and a -periodic orbit near if and only if .(v) The system (3) has exactly a unique 1-periodic orbit near if and only if satisfies .

Proof. Through taking a proper scale transformation, such as , we can treat as a small parameter in (28). According to and the implicit equation theorem, near , (28) can define a -dimensional surface , such that if , (28) has exactly two negative small solutions and .
If , then (28) has exactly a unique negative small solution .
If , that is , then (28) has exactly two negative small solutions .
If , then (28) has no negative small solution.
If , then (28) has exactly two negative small solutions and , moreover, .
The proof is complete.

Similarly, we can define the corresponding and , and the corresponding and , to obtain the following theorem.

Theorem 6. Suppose (H1)~(H3) and (H5) are fulfilled, , . If , then, in the neighborhood of , there exists two -dimensional surfaces and such that for sufficiently small , the following conclusions hold.(i) The system (3) has a unique 2-fold 1-periodic orbit near if and only if .(ii) The system (3) has no -homoclinic loop and 1-periodic orbit near if and only if satisfies .(iii) The system (3) has exactly two 1-periodic orbits near if and only if satisfies .(iv) The system (3) has exactly a unique -homoclinic loop and 1-periodic orbit near if and only if .(v) The system (3) has exactly a 1-periodic orbit near if and only if satisfies .

is called 2-fold 1-periodic orbit bifurcation surface, is called 1-homoclinic loop bifurcation surface Figure 2.

582820.fig.002
Figure 2

Now, we consider the nonexistence of -homoclinic loop and -periodic orbit, where . We may assume that .

We rewrite the time from and as , . Suppose , , let be the time from to , . Similar to the previous discussion, we can get its associated successor function as follows:

Denote . The equation always has a unique solution ,  ,  ,  , substituting it into , we get If (32) has solution , then (32) is turned to

If , the above formula defines a -dimensional surface , now the 2-homoclinic loop is the 1-homoclinic loop.

If (32) has the solution , , then (32) is turned to

Easily, the necessary condition that (34) has solutions , is , , that is, is twisted. (32) is turned to

The above formula defines -dimensional surface , such that when and , the system (3) has a unique 2-homoclinic loop.

If (32) has the solution , , then (32) turns to

Let and be the left and right of the above formula, respectively, then and are tangent at some point if and only if (36) and the following formulas are fulfilled.

Thus, we can get

Obviously, the necessary condition that (38) has solutions is and , that is, is twisted. At present, satisfies . Substituting (38) into (37), we get

Suppose the surface defined by (35) is . We notice that every 1-periodic orbit also corresponds to a solution of (32). Thus, we have the following theorem.

Theorem 7. Suppose (H1)~(H3) and (H5) are fulfilled, if , , then system (3) has no any -homoclinic loop and -periodic orbit in the small neighborhood of   for sufficiently small.

7. Stability

Now, we consider the stability of homoclinic loop .

According to , if , and , we can get the following by (19)

By (17), we get , , substituting into (40), we get

According to (11), we know , , , , , . And by , , , we can get , . Meanwhile, we make Poincaré map restrict at the half transversal section , makes the transversal line maps to the segment approximately, where is called the shrinkage (expansion) rate. So, when , we can get for , . Hence, we have the following.

Theorem 8. If , homoclinic loop is weak stability, and has a -dimensional stable manifold and a -dimensional instable manifold, If , homoclinic loop is weak instability, and has a -dimensional stable manifold and a -dimensional instable manifold.
If and , similar to above, we can get not difficulty. Thus, we have the following.

Theorem 9. If , homoclinic loop is weak stable, and has a -dimensional stable manifold and a -dimensional instable manifold. If , homoclinic loop is weak instable, and has a -dimensional stable manifold and a -dimensional instable manifold.

Besides, by the above discussion, we can get the following.

Theorem 10. The homoclinic loop or the periodic orbit of the perturbed system have the same stability with the homoclinic loop of the unperturbed system.

8. Example

Now, suppose the system (3) is 2-dimensional, , . We consider the stability of the homoclinic orbit under the resonance case. Denote , , , We have the following two lemmas.

Lemma 11. is convergent, and .

Proof. In fact, from [13], there always exist a transformation coordinates, such that the system (3) has the following form in a small neighborhood of the origin: where , is . So, in , we have . Thus, in , if , .
The proof is complete.

Remark 12. Because the divergence integration is the invariant under the transformation (refer [4, 16]), so the function and of the divergence integration can be thought of the original forms of (3).

Lemma 13. .

Proof. According to , , we get , so, . And by the Liouville formula, we have , therefore, , and by , we get .
The proof is complete.

Combined with the Theorem 9, we have the following.

Theorem 14. If , the homoclinic orbit is stable; If , the homoclinic orbit is instable.

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 (no. 10671069) and the Applied Mathematics Enhancement Program of Linyi University.

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