Abstract and Applied Analysis

Volume 2013 (2013), Article ID 267826, 7 pages

http://dx.doi.org/10.1155/2013/267826

Research Article

Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

Tiansi Zhang,¹ Xiaoxin Huang,¹ and Deming Zhu²

¹College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

²Department of Mathematics, East China Normal University, Shanghai 200062, China

Received 12 September 2013; Accepted 19 November 2013

Academic Editor: Svatoslav Staněk

Copyright © 2013 Tiansi Zhang 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.

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.

1. Introduction and Hypotheses

The study of homoclinic flip bifurcations is comprehensively developed from the last two decades with the beginning work of Yanagida (1987) for homoclinic-doubling bifurcations. Generally there exist two kinds of homoclinic flips, namely the orbit flips and the inclination flips corresponding to nonprincipal homoclinic orbits or critically twisted homoclinic orbits, respectively. Kisaka et al. in [1, 2] and Naudot in [3] studied some cases of codimension two inclination flips; Morales and Pacifico in [4] and Naudot in [5] considered the orbit flips cases, while Homburg and Krauskopf in [6] proposed several unfoldings of the resonant homoclinic flip bifurcations around the central codimension-three point (the organizing centre) in parameter space to study the qualitative structure of bifurcation curves on a sphere and also that of Oldeman et al. in [7] by a numerical investigation with some software into these bifurcations in a specific three-dimensional vector field.

Recently, Zhang et al. in [8–10] studied a kind of multiple flips homoclinic resonant bifurcation and got the existence of some saddle-node bifurcations and homoclinic-doubling bifurcations. Meanwhile Geng et al. in [11], Lu et al. in [12], and Liu in [13] discussed, respectively, a heterodimensional cycle flip or accompanied by transcritical bifurcation; they found the double and triple periodic orbit bifurcations and gave also some coexistence conditions for homoclinic orbits and periodic orbits.

As mentioned in [6, 7], due to the break of three genericity conditions, there are many complicated homoclinic flips cases to study. In this paper, we confine our attention to a principal eigenvalue resonance of one orbit flip and one inclination flip homoclinic bifurcation. Compared with the above-mentioned work, our subject is very challenging and difficult because of the stronger degeneracy and the higher codimension. By constructing specifically a local active coordinate in a small tubular neighborhood of homoclinic orbit, we establish a regular map and then combine it with a singular map defined by the approximation solutions of system to build Poincaré return map (see also [14]). We obtain the existence of several 1-periodic orbit, 1-homoclinic orbit, and double 1-periodic orbits, as well as some bifurcation surfaces with the analysis of the bifurcation equation.

We first consider a system

and its unperturbed system

where , , , , , , and . Suppose that the linearization D has four simple real eigenvalues , , , and with . Accordingly, the stable manifold and the unstable manifold are both two-dimensional. Let and be the strong stable manifold and the strong unstable manifold of the saddle , respectively. Assume further that system (2) has a homoclinic orbit . Hereinafter, our arguments will spread based on the following three hypotheses.

 (H1) Resonance. , , where and . (H2) Orbit Flip. Define ; ; then and are unit eigenvectors corresponding to and , respectively, where (resp., ) is the tangent space of the corresponding manifold (resp., ) at the saddle . (H3) Inclination Flip. Let and be the unit eigenvectors corresponding to and , respectively, and

Remark 1. Hypothesis (H2) is called an orbit flip because homoclinic orbit trends from the weak unstable manifold toward the strong stable manifold. Hypothesis (H3) means an inclination flip for its equivalence to

2. Poincaré Return Map

This section treats mainly the establishment of Poincaré return map with two steps. To begin we first need to transform system (1) into a normal form in some neighborhood of the origin .

It is well known that there are always two and transformations successively, also by the stable (or unstable) manifold theorem in [15], to straighten the local manifolds and as and , respectively, (resp., ); see [8–10]. Notice that now and , where and . Thus, system (1) can be changed to a form in the neighborhood as follows:

where , , , , , and . , , , and are parameters depending on .

Owing to the above straightness of the invariant manifolds, it is easy to find some moment , such that and for some sufficiently small and with . Wherefore one can choose

as the cross-sections of at and , respectively. Let be the time going from to and the Silnikov time , with the help of the linear approximation solutions of system (5) (see [8–10]); we have thereby

which give explicitly the definition of the local transition map ; see Figure 1(a).

(a) : →

(b) : →

Figure 1: Transition maps.

In the following part we construct the map . Firstly consider the linear variational system

and its adjoint system

Lemma 2. There exists a fundamental solution matrix , , , of system (8) satisfying

where , , , and , . Moreover, , , , , for .

Proof. Notice that the tangent subspace is invariant and is straightened to be axis; it is possible to choose since . While for , it is because and corresponding to axis.

As to or , , one may refer to [8, 9] for the similar proof, but we omit the details here.

Remark 3. The matrix is a fundamental solution matrix of system (9). We denote it as ; then is bounded and tends to zero exponentially as due to and tends exponentially to infinity.

In fact can be regarded as a new local coordinate system along . So we may make a coordinate change as

where . Note that the new should satisfy system (1); that is,

An asymptotic expansion with respect to shows that

Via integrating both sides from to of this equation, one can finally obtain

where are the Melnikov vectors. And further .

Equation (14) defines exactly the map ; under the new coordinate system; see Figure 1(b).

In order to combine and into the Poincaré return map, we still need to establish a relationship between the original and the new coordinate systems. In doing so, recall that ; then by taking time and , respectively, together with ,,,, ,,, and ,,,, ,,, , we obtain immediately the following formulas:

With all of the above, the Poincaré return map is given as . Therefore, the associated successor function , , , , is

3. Bifurcation Results

From the definition of the Silnikov time , we know that a solution with of (16) or equivalently corresponds to a periodic orbit near and a solution with of (16) or equivalently corresponds to a homoclinic orbit near . It is enough to study the solutions of the successor function for bifurcation analysis. Consider for concision that we omit the dependence on parameter from now on in the exponent notation.

From and , there are

Put and into ; we have

which is the bifurcation equation. Furthermore, for ,

where . Implicit function theorem reveals that (16) has a unique solution as

with , , and . It means that system (1) has a unique periodic orbit as or a unique homoclinic orbit as , and they cannot coexist.

Theorem 4. Suppose that and are true; then system (1) has a unique 1-periodic orbit near for or has a unique 1-homoclinic orbit near as , and they do not coexist.

Proof. Clearly has a small positive solution ., for , and has a zero solution for which is a codimension-one hypersurface.

In the following part we restrict our attention on the case for . Define

where .

In order to well solve (18), we rewrite it into two parts, namely, a line and a curve with respect to :

Then there are firstly the following conclusions based on an analysis of the relative position of the line and the curve .

Lemma 5. Suppose that , , and hold; then has a unique small positive solution for , where .

Proof. It is clear that

Therefore, the line intersects the curve at a unique point for . Notice that , so .

Theorem 6. Suppose that , , and hold; then system (1) has exactly a unique (resp., not any) 1-periodic orbit for (resp., ).

Proof. From Lemma 5, we know that has exactly a unique small positive solution for which corresponds exactly to a 1-periodic orbit of system (1). Moreover, does not have any small positive solutions for .

Theorem 7. Suppose that , , and hold; then, for and Rank , system (1) has a unique double 1-periodic orbit near on the bifurcation surface

which has a normal vector at . The corresponding double positive zero point is

as (see Figure 2(a)).

(a) Double 1-periodic orbit

(b) Two 1-periodic orbits

(c) No 1-periodic orbit

Figure 2: Rank and .

Proof. We know that the existence of a double 1-periodic orbit corresponds to the equations , , and , that is,

having solutions. Indeed, the second equation of (26) permits the double positive zero point as . Putting it into the first equation of (26), there is

Then exists for .

From the above proof, we see that, when and , the line has a positive slope lying under the curve when , so if increases, the line must intersect the curve at two sufficiently small positive points, which can be equal to the existence of two 1-periodic orbits of system (1). For , , and , the arguments are similar. So we have immediately a complement of Theorem 7.

Corollary 8. Assume that the hypotheses of Theorem 7 are valid, system (1) then has two (resp., not any) 1-periodic orbits near when lies on the side of which points to the direction (resp., in the opposite direction) (see Figures 2(b) and 2(c)).

As Melnikov functions generally play an important role in bifurcation study, the following theorem shows also the existence of some double 1-periodic orbits relying on the investigation of for .

Theorem 9. Suppose , , and are valid; then the following applies.

(1)For or , system (1) has exactly one 1-homoclinic orbit and one (resp., not any) 1-periodic orbit near and they (resp., do not) coexist as (resp., ).(2)For , system (1) has exactly one (resp., not any) 1-periodic orbit near as (resp., ). system (1) has a unique double 1-periodic orbit near as and . Accordingly, the double 1-periodic orbit bifurcation surface is with a normal vector at , and it may generate two 1-periodic orbits when lies in the direction of and no such a 1-periodic orbit in the opposite direction.(3)For or , system (1) has only one (resp. not any) 1-periodic orbit near as (resp., ).(4)For , system (1) does not have any 1-periodic orbit near .(5)For , system (1) has only one 1-homoclinic orbit near .

Proof. When or , has two solutions and . for which correspond to a 1-periodic orbit, and a 1-homoclinic orbit respectively. Thus, is true.

The result of the cases is exactly the same as that of Theorem 7.

If or , has only a positive solution . for , which means system (1) has a 1-periodic orbit. Then is valid.

In case of , it is clear that does not have any small nonnegative solutions, so system (1) does not have, any 1-homoclinic orbits or 1-periodic orbits.

The last conclusion is obvious. Thereby, the proof is complete.

Remark 10. Notice that, in Theorem 9   and , has a solution , which means that system (1) has a codimension-1 1-homoclinic orbit (see Figure 3(a)), so the existing homoclinic orbit is no longer orbit flip for . But if , an orbit flip homoclinic orbit could still exist, where is given by ; see Figure 3(b).

(a) ,

(b) ,

Figure 3: 1-homoclinic orbit (1-H) and (1-OH).

Remark 11. There still exist some double 1-periodic orbits or triple 1-periodic orbits for the case and ; one may pursue the similar process to discuss, so we leave it here.

Acknowledgment

The project is supported by the National Natural Science Foundation of China (no. 11126097).

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