Abstract and Applied Analysis

Volume 2012 (2012), Article ID 678252, 12 pages

doi:10.1155/2012/678252

Research Article

Bifurcations of a Homoclinic Orbit to Saddle-Center in Reversible Systems

Zhiqin Qiao¹ and Yancong Xu²

¹Department of Mathematics, North University of China, Taiyuan 030051, China

²Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Received 24 July 2012; Accepted 13 October 2012

Academic Editor: Xinan Hao

Copyright © 2012 Zhiqin Qiao and Yancong Xu. 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.

The bifurcations near a primary homoclinic orbit to a saddle-center are investigated in a 4-dimensional reversible system. By establishing a new kind of local moving frame along the primary homoclinic orbit and using the Melnikov functions, the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding codimension 1 or codimension 3 surfaces, are obtained.

1. Introduction

In recent years, there are much interest in the phenomenon of homoclinics and heteroclinics in reversible dynamical systems because of their extensive applications in mechanics, fluids, and optics [1–10]. For example, Klaus and Knobloch [4] considered a homoclinic orbit to saddle-center with two-parameter families of non-Hamiltonian reversible vector fields by Lin's method. They derived the occurrence of -homoclinic orbits to the center manifold. Liu et al. [6] studied a singular perturbation system with action-angle variable and the unperturbed system was assumed to possess a saddle-center equilibrium in a general system without reversible or Hamiltonian structure. Mielke et al. [7] investigated bifurcations of homoclinic orbit to saddle-center in 4-dimensional reversible Hamiltonian systems. By using the Poincaré map and a special normal form, they detected the existence of -homoclinic orbits to the equilibrium, -periodic orbits, and chaotic behavior near the primary homoclinic orbit. As for purely Hamiltonian system in , similar results are known from Koltsova and Lerman [2, 5]. In all of these papers the underlying Hamiltonian structure was heavily considered. Especially it was used to detect multiround orbits. Note that, the Hamiltonian can make the dynamics constrict in three-dimensional manifold in a zero level set. In this paper, we use the method originated by Zhu [9], by constructing a local moving coordinate system and Poincaré map near the primary homoclinic orbit, the existence of transversal homoclinic orbits and periodic orbits bifurcated from the primary homoclinic orbit are obtained in a 4-dimensional reversible system. It is worth to mention that a new kind of moving coordinates is introduced firstly in our paper in order to simplify and facilitate the reversible system.

The remainder of this paper is organized as follows. Section 2 contains the assumptions for the perturbed and unperturbed system. The local coordinate moving frame, cross sections, and Poincaré map are set up in Section 3. Finally, we obtain the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding surfaces with different condimensions in Sections 4 and 5.

2. The General Setup

Consider the following system

and the corresponding unperturbed system

where , , and are , and is a parameter. Also, we need the following assumptions.

System (2.1) is reversible with respect to the linear involution such that , for all and .

Note that, throughout the paper, we will denote R-symmetric orbit as symmetric orbit.

The origin is a saddle-center equilibrium of (2.2), . More precisely, the Jacobian matrix has a pair of purely imaginary eigenvalues and two nonzero real eigenvalues, that is, with .

System (2.2) has a symmetric homoclinic orbit , where .

Note that, is a symmetric equilibrium, and the eigenvalues of the Jacobian matrix are symmetric with respect to the imaginary axis. Thus, assumption describes a scenario that is structurally stable. Furthermore, the saddle-center has one-dimensional stable manifold and one-dimensional unstable manifolds (abbr. and as ), and a two-dimensional center manifold (abbr. as ) for close to . All of them are . Confined to , is a center. The reversibility implies that and , and hence, the homoclinic orbit is symmetric, that is, .

3. Local Moving Frame and Poincaré Map

Suppose the neighborhood of is small enough, we can firstly straighten the center-stable manifold, the center-unstable manifold, subsequently, then the stable manifold and the unstable manifold in by using the method introduced in Zhu [9]. According to the invariance and symmetry of these manifolds, we can deduce that system (2.1) has the following form in :

where , , , the system is , and the corresponding involution acts as .

In fact, by a linear transformation, system (2.1) takes the form in a small neighborhood of as follows:

and . By the invariant manifold theorem, we know that there exist a local center-stable manifold , a local center-unstable manifold and .

By the straightening coordinate transformation which is similar to that of [1, 6, 9], now we straighten the local manifolds and , such that .

Notice that the invariance of and implies the local invariance of and , respectively, which produces that, in ,

Now the system is and still reversible. By using a similar procedure to straighten the local stable manifold and unstable manifold , and the invariance and symmetry of these two local manifolds (that means the transformation is also symmetric), we get system (3.1). Clearly, corresponding to system (3.1), the center manifold is locally in the - plane, and the stable manifold (resp., unstable manifold ) is locally the -axis (resp., -axis) when they are confined in .

Define , for , where is small enough such that .

Consider the linear variational system

and its adjoint system

Based on the invariance and symmetry of manifolds and , it is easy to know that system (3.4) has a fundamental solution matrix satisfying , , and

Actually, in the resulting coordinates, and are -axis and -axis, respectively, combining with the symmetry, it follows that

On the other hand, in a small tubular neighborhood of the homoclinic loop , the center-unstable manifold (resp., center-stable manifold ) can be foliated into a family of leaves, each is a -dimensional surface and asymptotic to as the base point tends to as (resp., ). Notice that the limit of the linearization (3.4) of system (2.2) with respect to as is

When confined on , it becomes the following subsystem

which is reversible with the involution . Obviously, for any and , , is a solution of (3.9), which defines a closed orbit on . More precisely, its tangent vectors , and its normal vectors , are the solutions of (3.9). Choose some appropriate and such that for some , then . Based on the reversibility, we have (see Figure 1 for details). Thus, if we take solutions and in satisfying , then, restricted to the - plane, is the unit tangent direction of the closed orbit at . In addition, the restriction of is its unit exterior normal direction at the same point. By the reversibility, it is easy to obtain , .

Figure 1: The geometry of vectors on the center manifold.

Finally, if we choose a solution with , then the symmetry says that .

Therefore, we have demonstrated the existence of the fundamental matrix with the specified properties.

Remark 3.1. In the following, we will regard as a moving coordinate in a small tubular neighborhood of . This new kind of moving frame is firstly introduced for the homoclinic orbit to a saddle-center, which is the extension of the corresponding coordinates built in [1, 6, 9] for the homoclinic orbit to a saddle. The explicit advantage is that, these coordinate vectors inherit and exhibit the geometrical and dynamical properties of those invariant manifolds. As mentioned above, they will greatly simplify the original reversible system.

Let and be the cross sections of at and , respectively.

Now we turn to seek the new coordinates of and (see Figure 2 for details) under the transformation , where . Take

which are solved by

Figure 2: Setup of cross sections and Poincaré map.

Putting the transformation into (2.1), we get

That implies , which is . Then multiplying the resulting equation by and integrating it on both sides from to , we have the regular map :

where are the Melnikov functions.

For conciseness, we will denote . Now we consider the local map induced by the flow (3.1) in , where

Let be the flying time from to , by variation of constants formula, we can get the following expression:

where , , , and for , for .

Denote , then we get defined by

Combining the maps (3.13) and (3.16), we obtain the Poincaré map defined as

4. Existence of 1-Homoclinic Orbit and 1-Periodic Orbit

Let be the displacement function. Based on (3.11), (3.17), and the new coordinate of , we see the small zero point of will satisfy the following equations:

Due to the coordinate transformations introduced in at the beginning of Section 3, the unstable manifold and the stable manifold are locally -axis and -axis, respectively, so it is evident that, near , system (2.1) has a symmetric -homoclinic orbit to if and only if (4.1) have a solution with , and system (2.1) has an -homoclinic orbit to a periodic orbit on the center manifold and an -periodic orbit if and only if (4.1) have a solution with , and , , respectively. Clearly, system (4.1) is in as , , and in , and has a solution as , thus we can rescale , , , such that system (4.1) is reformulated as

where depends on and .

Now the following results are verified directly by the implicit function theorem.

Theorem 4.1. Assume ()–() are satisfied, and for and , then, for any given , and small enough, there exist a single-parameter family of codimension surface and a four-parameter family of codimension surface near such that , for , , , and system (2.1) has a symmetric -homoclinic orbit to as , a -periodic orbit with approximate period as , and a -homoclinic orbit to a periodic orbit on the center manifold with as , , . Moreover, these codimension surfaces are either coincident, or tangent to each other at , and they have a common -dimensional normal space spanned by ,, and as .

In order to study the existence of the -periodic orbit bifurcation surface with lower codimension, we reconsider (4.1). Let , and be the vector defined by the left hand side of the last two equations of (4.1), then, by using

it follows that, for so small that , , we have for . Therefore, there is a unique pair which solves the last two equations as and is small enough.

Substituting this solution into the first equation of (4.1), and using for , it produces

where the term depends on and .

At this stage, we have actually demonstrated the following existence theorem.

Theorem 4.2. Suppose ()–() are satisfied, , as and , then for , small enough, , and , there exists a -parameter (resp., -parameter) family of codimension surface (resp., ) near satisfying , such that system (2.1) has one -periodic orbit (resp., -homoclinic orbit ) near as (resp., ), and has an approximate period . In addition, if for any , then system (2.1) has not any -periodic orbit near .

Remark 4.3. When confined in a neighborhood of the local center-manifold, the ()-coordinates of -periodic or -homoclinic orbit guaranteed by Theorem 4.1 as or satisfying , whereas, these coordinates of the corresponding orbits given in Theorem 4.2 as or have the scale in case and .

5. Existence of Symmetric 1-Periodic Orbit

In this section, we turn to seek the existence of symmetric -periodic orbit. Note that, a -periodic orbit near is symmetric if and only if , equivalently and should satisfy

Due to (3.15), it is equivalent to

Using (5.2) and rescaling , , for , , system (4.1) reads as

Solving the second equation of (5.3), we have

On the other hand, and are symmetric if and only if their middle point lies on the plane , that is, (5.1) is equivalent to

where

and for , , for , for .

Explicitly, (5.5) can be formulated as

Due to (5.4) and , , we can uniquely solve the second equation in (5.7) by

as , , and , small enough.

Up to now, there are three equations

left which should be fulfilled, where and are given by (5.4) and (5.8).

Applying the implicit function theorem to (5.9), where depends on and , we derive the following result.

Theorem 5.1. Suppose that ()–() hold, , and

as , there exist , small enough, and a -parameter family of codimension surface near , such that, system (2.1) has a symmetric -periodic orbit near with approximate period as for satisfying , , , . Moreover, if there is no satisfying

then system (2.1) has no symmetric -periodic orbit in the small neighborhood of .

Remark 5.2. Similarly, if we take in (3.15), then (i.e., condition (5.1)), it means that there is a symmetric 1-homoclinic orbit to if and only if in Section 4.

Remark 5.3. In Theorem 4.2 (resp., Theorem 5.1), the geometric meaning of is that, confined in , the - component of the above -periodic orbit bifurcated from makes circle (resp., ) times around the saddle-center.

Remark 5.4. From , and the constitution of and , it is easy to know that the necessary condition for is .

Acknowledgments

This paper is supported by NSF of China (11101370, 11126265, 11171085), the Research Fund for the Doctoral Program of Higher Education (20111420120006), and Zhejiang Provincial NSF of China (Y6100081).

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