ABSTRACT

One orbit flip and two inclination flips bifurcation is considered with resonant principal eigenvalues. We introduce a local active coordinate system to establish bifurcation equation and obtain the conditions when the original homoclinic orbit is kept or broken. We also prove the existence and the existence regions of double 1-periodic orbit bifurcation. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately, and are well located.

1. Introduction

Homoclinic bifurcations have been comprehensively investigated from the initial work of Silnikov in [1] who gave a detailed study of a system which permits an orbit homoclinic to a saddle-focus. After that many flips cases attract researcher’s interests, including resonant eigenvalues case in [2], orbit flips in [3,4], inclination flips in [5-7], and also resonant homoclinic flips in [8-11]. In these cases homoclinic-doubling bifurcation has been expensively studied, which is a codimension-two transition from an n-homoclinic to a 2n-homoclinic orbit. Some applications of these cases may be referred to a model for electro-chemical oscillators, the FitzHugh-Nagumo nerveaxon equations [12], a Shimitzu-Morioka equation for convection instabilities [13], and a Hodgkin-Huxley model of thermally sensitive neurons [14], etc.

More recently, the flip of heterodimensional cycles or accompanied by transcritical bifurcation is got attention, see [15-17], the double and triple periodic orbit bifurcation are proved to exist, and also some coexistence conditions for the homoclinic orbit and the periodic orbit. But the research is not concerned with multiple flips. While multiple cases may have more complicated bifurcation behaviors and even chaos, it is necessary to give a deep study. This paper produces mainly a theoretical study of homoclinic bifurcation with one orbit flip and two inclination flips, which can take place at least in a fourdimensional system. Compared with the above work mentioned, our problem has higher codimension with resonant, and we get not only the existence of 1-periodic orbit, 1-homoclinic orbit, and double periodic orbit, but also the -homoclinic orbit and their corresponding bifurcation surfaces.

In the present context, we consider the following system

(1.1)

and its unperturbed system

(1.2)

where .

Hypothesis

We assume that system (1.2) has a homoclinic orbit to an equilibrium, which is hyperbolic and has two negative and two positive eigenvalues, denoted by, and additionally. Set (resp.) and (resp.) the stable (resp. strong stable) manifold and unstable (resp. strong unstable) manifold of the equilibrium, respectively. Now we further make three assumptions:

(H1) (Resonance) for, where

and.

(H2) (Orbit flip) Define,

, then 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) (Inclination flips) Denote by and the unit eigenvectors corresponding to and respectively, let

The paper is organized as follows. In Section 2 we will construct the Poincaré map by the method used in [18] to get the associated successor function. In Section 3, we first establish bifurcation equation. Then a delicate study shows our main results about the existence of double 1-periodic orbit, 1-homoclinic orbit and also -homoclinic orbit. The last section gives a conclusion of the work.

2. Two Normal Forms and Successor Function

From the above hypotheses, the normal form theory provides a system as follows after four successive to transformations in (see [10,11,18])

(2.1)

with the assumption

(H4).

Indeed we have

where

and

and

and

and are parameters depending on. Notice that we have straightened the corresponding invariant manifolds. So it is possible to choose some moment, such that and, where is small enough and

.

Now we turn to consider the linear variational system and its adjoint system

(2.2)

(2.3)

First we introduce a lemma, see [10,11]

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

where

and, and.

Remark 2.1 The matrix is a fundamental solution matrix of system (2.3), denote by

then

is bounded and tends to zero exponentially as due to and tends exponentially to infinity.

Let

(2.4)

where. We can well regard

as a new local coordinate system along, and choose

as the cross sections of at and respectively. Under the transformation of (2.4), system (1.1) becomes

A simple integrating of both sides from to of the above equation, we further achieve

(2.5)

where

are the Melnikov vectors (see [18]).

Lemma 2.2

.

Actually a regular map is given by (2.5) as (see Figure 1(a))

But this map is established in the new coordinate system, so we should look for the relationship between two coordinate systems. Set

and

Take respectively in (2.4), we have

(2.6)

(2.7)

and

(2.8)

Next, we start to set up a singular map

(see Figure 1(b)) induced by the solutions of system (2.1) in the neighborhood, for example

where is the time going from to. Denote the Silnikov time, then there is

.

Similarly, there are

(2.9)

With Equations (2.6)-(2.9), Equation (2.5) well defines the Poincaré map,

The above fact enables one to achieve the associated successor function as follows:

(2.10)

3. Main Results

To begin the bifurcation study, and first give

Substitute them into, we get

(3.1)

this is the bifurcation equation. Here we have omitted the parameter in and, and replaced the exponent by one owing to (H1) for concision.

Set, we find that, when,

The implicit function theorem reveals that has a unique solution

satisfying So system (1.1) has a unique periodic orbit as or a unique homoclinic orbit as, and they do not coexist. Furthermore, has explicitly a sufficiently small positive solution if. On the other hand, it has a solution when, so we have Theorem 3.1 Suppose that and hold, then system (1.1) has at most one 1-periodic orbit or one 1-homoclinic orbit in the neighborhood of. Moreover an 1-periodic orbit exists (resp. does not exist) as in the region defined by (resp.) and an 1-homoclinic orbit exists as, but they do not coexist.

In the following stage, we try to look for bifurcations according to the case for.

To begin with we divide (3.1) into two parts:

Therefore, where

is a line and is a curve with according to the variable.

Theorem 3.2 Suppose that and, system (1.1) then has a unique double 1-periodic orbit near, and two (resp. not any) 1-periodic orbits near when lies on the side of which points to the direction (resp. in the opposite direction of). The corresponding double 1-periodic orbit bifurcation surface is

with the normal vector at.

Proof Consider equations

and, that is,

(3.2)

The second equation permits a solution

as

.

Substituting it into the first equation of (3.2), we obtain the tangency condition, which corresponds to the existence of the double periodic orbit bifurcation surface situated in the region and. Notice that, when the tangency takes place, the line lies under the curve. So if increases (resp. decreases), the line must intersects the curve at two (resp. no) sufficiently small positive points. Now the proof is complete.

Theorem 3.3 Suppose that and are true, then there exists two codimension-one hypersurfaces

and

such that System (1.1) has only one 1-homoclinic orbit near as and;

System (1.1) has only one 1-periodic orbit near as and;

System (1.1) has exactly one 1-homoclinic orbit and one 1-periodic orbit near as and ;

System (1.1) has not any 1-periodic orbit or 1-homoclinic orbit as and.

Proof When, we have at once

and, therefore

has always two nonnegative solutions and

for or has only a zero solution

for. If, there is, on the contrary, but, apparently the line is horizontal. So has a solution

if and only if. The proof is complete.

From the above proof, we see that if the line has a small positive section with the - axis or small positive slope, then there exists a small positive such that. Thus the following corollary is valid, which is a complement of Theorem 3.2.

Corollary 3.4 Assume that the hypotheses of Theorem 3.2 are valid, system (1.1) then has a unique 1-periodic orbit near as is situated in the region defined by

and or, and; has not any 1-periodic orbit as and.

Notice that in Theorem 3.3, system (1.1) has a codimension-1 1-homoclinic orbit, see Figure 2(a), that is the existing homoclinic orbit has no longer orbit flip. But an orbit flip homoclinic orbit could still exist if

see Figure 2(b).

Corollary 3.5 Suppose that and hold, system (1.1) has a codimension-2 orbit-flip homoclinic orbit as

.

Now we turn to study the homoclinic doubling bifurcations. To begin with we look for the 2-homoclinic orbit and 2-periodic orbit bifurcation surfaces. Reset and be the time going from to

and from to respectively, and

.

Then recall the process of the establishment of (2.10), similarly we may get the associated second returning successor function expressed as

:

Eliminating again and from and, and assuming

we obtain

(3.3)

(3.4)

We know that a 2-homoclinic orbit corresponds to the solution and or and of (3.3) and (3.4), that means an orbit returns once nearby the singular point in limit time and twice in limitless time. So it is sufficient to seek the small solutions of and by symmetry of. Therefore

(3.5)

(3.6)

Clearly (3.5) yields

for, and sufficiently small. With this, Equation (3.6) determines a 2-homoclinic orbit bifurcation surface

for and, which has a normal vector at.

Continually, differentiating both sides of (3.3) and (3.4) with respect to and for, we obtain

In the region defined by

andthe 2-homoclinic orbit bifurcation surface is simplified to be

.

Accordingly

.

Then one may derive

which informs that increases (resp. decreases) as moves along the direction (resp. the opposite direction) such that a -periodic orbit bifurcates from the -homoclinic orbit as leaves for the side pointed by.

Notice that confined on the surface, (3.3) and (3.4) has a unique positive solution, meanwhile Theorem 3.3 indicates exactly the existence of one 1-periodic orbit when for, so there does not exist any 2-periodic orbit when is near. Therefore in the region bounded by the surfaces to, there must exist another bifurcation surface which merges the 1-periodic orbit and the 2-periodic orbit into a new 1-periodic orbit with the different stability from the original one. We call this surface the period-doubling bifurcation surface and denote it by.

The above reasonings can repeat itself many times to find the -homoclinic orbit bifurcation surface

in the same region of space and simultaneously the presence of period-doubling bifurcation surface of -periodic orbit.

In short, we conclude that:

Theorem 3.6 Suppose that

and hold, then for

and, there exists a -homoclinic orbit bifurcation surface with the normal vector at and the period-doubling bifurcation surface of -periodic orbit in the small neighborhood of the origin of space. Moreover system (1.1) has exactly a -homoclinic orbit as and a -periodic orbit as moves away to the side of pointing to the direction and none on the other side.

To well illustrate our results, a bifurcation diagram is drawn in Figure 3, where represents a -periodic orbit.

4. Conclusion

Homoclinic orbits generically occur as a codimensionone phenomenon, while if the genericity conditions are

broken, some high codimension instance including the resonant and flips cases, concomitant usually with chaotic behavior, may take place. Homburg and Oldeman studied two kinds of resonant homoclinic flips in [8,9] with unfolding techniques and numerical methods respectively. Zhang in [10,11] continued to research on these problems and gave some theoretical proofs of the existence of -periodic orbit and -homoclinic orbit and also their existence regions via the method initially established in [18]. Besides these the flip heterodimensional cycles have also attracted attentions nowadays, see [16]. In this paper, we extend the method to fit a higher codimension case of 3 flips with resonant. With the delicate analysis, the existence of 1-periodic orbit, 1-homoclinic orbit, and double periodic orbit are proven and also the -homoclinic orbit and their corresponding bifurcation surfaces. With the work, we find the extensive existence of the double periodic orbit bifurcation and the homoclinic-doubling bifurcation, which efficiently advance the development of the flips homoclinic study.

REFERENCES

NOTES

^*Project supported by National Natural Science Foundation of China (Grant: 11126097) and by SRF for ROCS, SEM.