The Scientific World Journal

Volume 2014 (2014), Article ID 716082, 8 pages

http://dx.doi.org/10.1155/2014/716082

Research Article

Bifurcations of Nontwisted Heteroclinic Loop with Resonant Eigenvalues

Yinlai Jin,¹ Xiaowei Zhu,1,2 Zheng Guo,¹ Han Xu,¹ Liqun Zhang,1,2 and Benyan Ding1,2

¹School of Science, Linyi University, Linyi, Shandong 276005, China

²School of Mathematics Science, Shandong Normal University, Jinan 250014, China

Received 21 January 2014; Accepted 17 March 2014; Published 16 April 2014

Academic Editors: M. Han, Z. Jin, and Y. Xia

Copyright © 2014 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 heteroclinic orbits to establish the local coordinate systems in the small tubular neighborhoods of the heteroclinic orbits, we study the bifurcation problems of nontwisted heteroclinic loop with resonant eigenvalues. The existence, numbers, and existence regions of 1-heteroclinic loop, 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits are obtained. Meanwhile, we give the corresponding bifurcation surfaces.

1. Introduction and Hypotheses

With the development of nonlinear science and the deep study of chaotic phenomenon, many experts have been interested in the research on the bifurcation problems of homoclinic and heteroclinic loops for high-dimensional nonlinear dynamical systems (see [1–5] and references therein). Compared with the bifurcations of homoclinic and heteroclinic loops for planer systems, the obtained results have been not rich because of the complexity and the lack of methods. In 1990, Chow et al. studied the codimension 2 homoclinic bifurcation under nondegenerate condition at resonant eigenvalues in [6]. The traditional methods to construct Poincaré maps were adopted to later research. In 1998, Zhu studied the bifurcation problems of nondegenerated homoclinic loops for high-dimensional system in [7] and the bifurcations of nondegenerated heteroclinic loops for high-dimensional system in [8]. The methods used in [7, 8] were by generalizing the Floquet method to build the local coordinate systems and Poincaré map. The papers [7, 8] used the inherent characteristic values to describe the bifurcation surfaces and phenomena so that the results had the practicability and maneuverability.

In [9], the authors studied the bifurcations of rough heteroclinic loops with two saddle point. In [10], Tian and Zhu studied the bifurcations of fine heteroclinic loops with two saddle points. The heteroclinic loops studied in [9, 10] were both nontwisted and with no resonant eigenvalues.

In this paper, we study the bifurcation problems of heteroclinic loop with resonant eigenvalues for high-dimensional system . By simplifying the method in [7, 8], we use the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits to establish the suitable local coordinate systems in the small tubular neighborhoods of the heteroclinic orbits. Then, we get the Poincaré maps and bifurcation equations by means of the improved method.

Consider the system

where , . We assume the following.

(H1) Hyperbolic assumption: are hyperbolic critical points of (1), . The stable manifold and the unstable manifold of are -dimensional and -dimensional, respectively. Moreover, and are the simple real eigenvalues of such that any other eigenvalue of satisfies either or , where and are some positive constants.

(H2) Nondegeneration: system (1) has a heteroclinic loop , where , , , . .

(H3) Strong inclination: , , where and are the strong unstable manifolds and the strong stable manifolds, respectively, is the generalized eigenspace corresponding to all the eigenvalues with larger real part than , and is the generalized eigenspace corresponding to all the eigenvalues with smaller real part than . Denote , where and are unit eigenvectors corresponding to and , respectively. Furthermore, , .

Now, we consider the following system:

where , , , , .

2. Local Coordinate Systems

Suppose that is a sufficiently small neighborhood of and (H1)(H3) hold; then, for small enough, there exists a transformation such that system (2) has the following form in , respectively:

where , , , , , means transposition, , , , and . Moreover, we suppose

where , , , .

Let ; we can select some times and such that , , where is small enough such that , , , .

Consider the linear variational system and its adjoint system

By [8, 9, 11–13], system (5) and (6) have exponential dichotomies in and . It follows from [4] that system (5) has a fundamental solution matrix , satisfying

where , , , , and , ; , ; , ; , for small enough.

Denote ; then is a fundamental solution matrix of (6). It is well known that is bounded and tends to zero exponentially as [8, 9, 12].

Denote . is nontwisted if and twisted if . In this paper, we study the bifurcation problems under the nontwisted condition.

Let , , , be the local coordinate systems along in the tubular neighborhood of .

3. Poincaré Maps and Bifurcation Equations

Denote , , , . Define the Poincaré sections of at and by

respectively, where is small enough such that , , (Figure 1).

Figure 1

Now, we set up the Poincaré map : , where : , : , , and . Here, is constructed from the flow of (2) in the small tubular neighborhood of ; is induced by the flow of the linear approximate system of (3) in the small neighborhood of .

Denote

where , .

By the expressions of and , we get that the relations of the two kinds coordinates of are , , and

First, we built the map , . We make a coordinate transformation , and substitute it into (2). By , and ; (2) becomes

Thus, we define , as

That is,

where , , , are called the Melnikov vectors.

By [7–9, 14], , , .

Second, we built the map , , , . Without loss of generality, we assume . Let be the flying time from to ; we say is Silnikov time. By (3) and neglecting the higher order terms, we have

Here, is called the Silnikov coordinate.

Last, it follows from (10)(14) that we get the expression of Poincaré map as follows:

Denote . By (10) and (15), we get the successor function as follows:

Let

Equation (17) is called the bifurcation equation.

4. The Preservations of Heteroclinic Orbits and Bifurcations of Homoclinic Loops

In this section, we discuss the nontwisted heteroclinic loop with resonant eigenvalues. Without loss of generality, we assume the following.

(H4) Resonant condition: , .

Let , , . By (H4) and the continuity of function, we get , , , , and , for .

Consider the solution of (17). Clearly, by the implicit function theorem, the equation has a unique solution , , . Substituting it into , we have

Equation (18) is equivalent to

Now, we study the existence of heteroclinic orbits and the -homoclinic bifurcations. Denote

Theorem 1. Suppose that hypotheses (H1)(H4) are valid, and rank ; then one has the following.

(i) There exists a -dimensional surface with a normal vector at such that (2) has a heteroclinic orbit joining and near if and only if , , where . Moreover, (2) has a -heteroclinic loop near if and only if and , where is -dimensional surface and . That is, heteroclinic loop is persistent.

(ii) There exists a -dimensional surface which is tangent to at as , such that (2) has a unique -homoclinic loop homoclinic to near as and . Meanwhile, there also exists a -dimensional surface which is tangent to at such that (2) has a unique -homoclinic loop homoclinic to near as and .

Proof. (i) If (19) has a solution , then, we have

If , then, there exists a -dimensional surface defined by (21) with a normal vector at such that the th equation of (19) has a solution as and . That is, is persistent.

If rank , then there exists a -dimensional surface such that (19) has a solution as and . That is, is persistent.

(ii) If (19) has a solution , , then, we have

If , , then, (22) defined a -dimensional surface with a normal vector at as , such that (19) has a solution , as and . That is, the system (2) has a -homoclinic orbit to near for and .

(iii) If (19) has a solution , , then, we have

If , , then, (23) defined a -dimensional surface with a normal vector at such that (19) has a solution , as and . That is, the system (2) has a -homoclinic orbit to near for and .

5. The Periodic Orbits Bifurcations

Next, we consider the bifurcation problems of -periodic orbits near . In other words, we study the solutions of (19) satisfying , . We assume the following.

(H5) Nontwisted condition: .

Obviously, if (H5) holds, we have

At first, we consider the bifurcations in . From (19), we get

Denote and are the left and right hands of (25), respectively; then, we have the following.

Lemma 2. Suppose that hypotheses (H1)(H5) are valid; then is tangent to at some point satisfying , if and only if , , , and

Proof. is tangent to at some point if and only if and ; that is,

So

Substituting (28) into the second expression of (27), we have that (26) holds.

Equation (28) means , and (26) means and .

If and are linearly independent, then there exists a -dimensional surface defined by (26) in the small neighborhood of . It is easy to know that is tangent to for . By (21), (22), and (26), we get

That is to say, is located in the open region between and . Thus, we have the following.

For , we define the following three open sections in : is bounded by and , is bounded by and , and is bounded by and , and they have nonempty intersection with .

For , we define the following two open sections in : is bounded by and and is bounded by and , and they have nonempty intersection with .

Now, we consider the nonnegative solutions of defined by (25). By , , we get , . From (22), holds for , .

By Lemma 2, it is easy to get the following conclusions.

(1)For , we have the following (see Figure 2). If , then has one small positive solution. If , then has one small positive solution and one zero solution. If , then has two small different positive solutions. If , then has one small twofold positive solution. If , then does not have any small nonnegative solutions.(2)For , we have the following. If , then has one small positive solution. If , then has one zero solution. If , then has not any small nonnegative solutions.

(a)

(b)

(c)

(d)

(e)

Figure 2

Thus, we have shown the following conclusions.

Theorem 3. Suppose that hypotheses (H1)(H5) are valid, , ; then the following conclusions are true.

(i)System (2) has one simple -periodic orbit near as .(ii)System (2) has one simple -periodic orbit and one -homoclinic loop homoclinic to near as .(iii)System (2) has two simple -periodic orbits near as .(iv)System (2) has a unique twofold -periodic orbit near as .(v)System (2) has not any -periodic orbits and -homoclinic loops near as .

Theorem 4. Suppose that hypotheses (H1)(H5) are valid, , ; then the following conclusions are true.

(i)System (2) has one simple -periodic orbit near as .(ii)System (2) has one -homoclinic loop homoclinic to near as .(iii)System (2) does not have any -periodic orbits and -homoclinic loops near as .

Next, we consider the bifurcations in . Similarly, from (19), we get

Denote and are the left and right hands of (30), respectively; then one has the following.

Lemma 5. Suppose that hypotheses (H1)(H5) are valid, then is tangent to at some point satisfying , if and only if , , and

Proof. is tangent to at some point if and only if and ; that is,

The solution of (32) is

Substituting (33) into the second expression of (32), we obtain (31).

Equation (33) holds means , , or , . But, if , , then, by (31), we get that will be tented to . Last, is obvious.

If and are linearly independent, then there exists a -dimensional surface defined by (31) in the small neighborhood of . It is easy to know that is tangent to at . By (21), (23), and (31), we get

That is, is located in the open region between and . Thus, we have the following.

For , we define the following two open sections: is bounded by and and is bounded by and , and they have nonempty intersection with .

For , we define the following three open sections: is bounded by and , is bounded by and , and is bounded by and , and they have nonempty intersection with .

Now, we consider the nonnegative solutions of defined by (30). By , , we get , . From (23), holds for , .

By Lemma 5, it is easy to get the following conclusions.

(1)For , we have the following.

 If , then has one small positive solution. If , then has one zero solution. If , then does not have any small nonnegative solutions.

(2)For , we have the following.

 If , then has one small positive solution. If , then has one small positive solution and one zero solution. If , then has two different positive solutions. If , then has one twofold positive solution. If , then does not have any small nonnegative solutions.

Thus, we have shown the following conclusions.

Theorem 6. Suppose that hypotheses (H1)(H5) are valid, , ; then the following conclusions are true.

(i)System (2) has one simple -periodic orbit near as .(ii)System (2) has one -homoclinic loop homoclinic to near as .(iii)System (2) does not have any -periodic orbits and -homoclinic loops near as .

Theorem 7. Suppose that hypotheses (H1)(H5) are valid, , ; then the following conclusions are true.

(i)System (2) has one simple -periodic orbit near as .(ii)System (2) has one simple -periodic orbit and one -homoclinic loop homoclinic to near as .(iii)System (2) has two simple -periodic orbits near as .(iv)System (2) has a unique twofold -periodic orbit near as .(v)System (2) does not have any -periodic orbits and -homoclinic loops near as .

Let be an open region that is bounded by and ; meanwhile, . Let be an open region that is bounded by and ; meanwhile, .

By (19), it is easy to see that (19) has an unique solution , for and does not have any nonnegative solutions for . So we get the following conclusions.

Theorem 8. Supposing that hypotheses (H1)(H5) are valid, then

(i)system (2) has one simple -periodic orbit near as ,(ii)system (2) has not any -periodic orbits, -homoclinic loops, and heteroclinic orbits near as .

Combining Theorems 1~8, we get the bifurcation figures (see Figures 3 and 4).

Figure 3

Figure 4

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), the Natural Science Foundation of Shandong Province (no. Y2007A17), and the Applied Mathematics Enhancement Program of Linyi University.

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