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