ABSTRACT

Homoclinic bifurcation with one orbit flip, two inclination flips and resonance in the tangent directions of homoclinic orbit is considered. By studying the associated successor functions constructed from a local active coordinate system, we prove the existence of double 1-periodic orbit, 1-homoclinic orbit, and also some coexistence conditions of 1-periodic orbit and 1-homoclinic orbit.

1. Introduction and Hypotheses

Flips homoclinic bifurcations are comprehensively investigated during the last decade (see [1-10]), which produce complicated bifurcations, such as the saddlenode bifurcations, the period-doubling bifurcations and the homoclinic-doubling bifurcations.

Recently, the flip of heterodimensional cycle or accompanied by transcritical bifurcation is discussed much (see [11-13]). The double and triple periodic orbit bifurcation are proved to exist, and also some coexistence conditions for homoclinic orbits and periodic orbits. But their research is not focused on multiple flips since it is a interesting problem and full of challenges due to the high codimension and complexity. In this paper, we develop a study of resonant homoclinic bifurcation with one orbit flip and two inclination flips, where the resonance takes place in the tangent direction of the homoclinic orbit. This is a codimension-4 problem, by using the local moving frame method established in [11,14,15], we get the existence of a double 1-periodic orbit, some 1-periodic orbits and 1-homoclinic orbits, and the coexistence conditions of 1-periodic orbits and 1-homoclinic orbits.

We consider the following two systems,

(1.1)

(1.2)

 

where .

Notice that system (1.2) is an unperturbed system of (1.1) and assume it has an orbit

homoclinic to the hyperbolic equilibrium, which has two negative and two positive eigenvalues, , and.

Hypotheses

Set (resp.) and (resp.) the stable (resp. strong stable) manifold and unstable (resp. strong unstable) manifold of the equilibrium, respectively. We suppose that

(H1) for, where and.

(H2) Definethen and are unit eigenvectors corresponding to and respectively, where is the tangent space of the corresponding manifold at the saddle, and the similar meaning for.

(H3) Denote by and the unit eigenvectors corresponding to and respectively, there are

Remark 1.1 Hypotheses (H1) is a resonant condition, while (H2) - (H3) mean the homoclinic orbit has one orbit flip and two inclinations flips.

The paper is organized as follows. In Section 2, we first transform system (1.1) into two normal forms, then construct a regular map in some neighborhood of the homoclinic orbit and a singular map in some neighborhood of the equilibrium respectively to establish the Poincaré map. In Section 3, we develop the bifurcation study through searching for solutions of the bifurcation equation. Finally a short conclusion about the flips bifurcation is given in Section 4.

2. Local Active Coordinate Frame and Poincaré Map

We first give two normal forms of system (1.1) and then construct the Poincaré map. Firstly system (1.1) can be transformed into the following form in some neighborhood of the origin due to the theory of invariance manifolds, (refer to [14,15])

(2.1)

where and. and are parameters depending on.

One may see that from (2.1), and are straightened locally to be the axes in neighborhood of, so it is possible to take some time large enough, such that and, where is small and

.

Now consider the linear variational system

(2.2)

and its adjoint system

(2.3)

Matrix theory shows that system (2.2) has a fundamental solution matrix and furthermore it can be chosen as follows (refer to [11,14-15])

Lemma 2.1 There exists a fundamental solution matrix of system (2.2) satisfying

where

and, and.

Obviously is a fundamental solution matrix of system (2.3), denote by

.

We here introduce a new coordinate and set

(2.4)

Naturally we can choose two cross sections of, see Figure 1,

Substitute (2.4) into (1.1), there is

Integrating both sides from to, we further have

(2.5)

where

.

Equation (2.5) defines indeed a map in some tube region near,

see Figure 1(a). If set

and

we can obtain the following expressions,

(2.6)

(2.7)

and

(2.8)

Using the flow of system (2.1) in the neighborhood, we can set up a map

defined as (see [14,15])

(2.9)

where is the Silnikov time and is the time going from to, see Figure 1(b).

From the above the Poincaré map

is obtained

Then the corresponding associated successor function is

(2.10)

Since is defined by the flying time from the point in to, obviously means is limited; means, which indicate the existence of a periodic orbit or a homoclinic orbit of system (1.1). So in the following section, we focus us on the solutions of (2.10).

3. Bifurcation Results

The last two equations in (2.10) give

Then from we get the bifurcation equation

(3.1)

Notice that we have put higher orders terms into and omitted the parameter in the eigenvalues for concision.

Define two functions as

Indeed here. By analysis of the curves and, one may immediately get the following statements.

Theorem 3.1 Suppose that, then in the region, system (1.1) has a unique 1-periodic orbit near; in the region, system (1.1) has not any 1-periodic orbit.

Proof Because

the curve has no inflexion point, so th e line and the curve must intersect at a unique point in the region, where

and

see Figure 2(a). Namely there exists a point such that

therefore system (1.1) has a unique 1-periodic orbit. On the contrary, there is not such a intersection point in the region

see Figure 2(b).

Theorem 3.2 Suppose that, then in the region, system (1.1) has a unique double 1- periodic orbit near located in the bifurcation surface.

Moreover when lies on the side of pointing to the (resp. opposite) direction, system (1.1) has two (resp. not any) 1-periodic orbits near.

Proof We know that the existence of a double 1-periodic orbit corresponds to a double solution of (3.1). According to the proof of Theorem 3.1, it is enough to search the tangent point of the curves and

, that is to solve

and, concretely,

(3.2)

Then the tangent point

as. Combining the first equation of (3.2) with the tangent point, we obtain the double periodic orbit bifurcation surface

in the region. At the same time, when, the line lies under the curve, see Figure 2(c), so if increases, the line must intersects the curve at two sufficiently small positive points, therefore system (1.1) undergos two 1-periodic orbits. Then the proof is complete.

Theorem 3.3 Suppose that, then system (1.1) has only one 1-homoclinic orbit near in the region; has only one 1-periodic orbit near in the region; has exactly one 1-homoclinic orbit and one 1-periodic orbit near in the region; has not any 1-periodic orbit or 1-homoclinic orbit in the region.

Proof When

,

has always two solutions and

or has only a zero solution for

.

While for

apparently the line is horizontal. So gives merely a solution

.

The last conclusion is obvious for

.

In the following, we study the case. Then (3.1) is

Similar to the proof of Theorem 3.1 and 3.3, we have Theorem 3.4 Suppose that, then in the region, system (1.1) has a unique 1-periodic orbit near; in the region, system (1.1) has not any 1-periodic orbit.

Proof Redefine

By studying the relationship between the curves and, it is easy to get the main ideas, see Figure 3. Here

and

.

Remark 3.1 For the case, system (1.1) has no longer double 1-periodic orbits and the double 1-periodic orbit bifurcation surfaces.

Theorem 3.5 Suppose that, then system (1.1) has only one 1-homoclinic orbit near in the region; has only one 1-periodic orbit near in the region; has not any 1-periodic orbit or 1-homoclinic orbit in the region.

Proof Notice that

has only a zero solution for

.

And the line is horizontal for

Thereby the conclusion is clear. We omit the details here.

4. Conclusion

The theoretical development of flip bifurcations indeed advanced much in recent years. More and more complicated cases with several flips or accompanied by transcritical bifurcation nowadays are discussed. This paper focuses on a kind of three flips homoclinic case with resonance and introduces an effective method to extend the study. By the analysis of the bifurcation equation, the existence of a double 1-periodic orbit, some 1-periodic orbits and 1-homoclinic orbits, and the coexistence conditions of 1-periodic orbits and 1-homoclinic orbits are given. From the study, one notice that different leading terms of the bifurcation equation may cause different bifurcation phenomena, so we can go further in the future work.

REFERENCES

[16]    NOTES

[17]    ^*Project supported by National Natural Science Foundation of China (Grant: 11126097) and by Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.