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The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

In recent years, there is a large literature concerning the bifurcation problems of homoclinic and heteroclinic loops in dynamical systems (see [

In this paper, on the one hand, using the method which was originally established in [

Motivated by these points, we will consider the following

We make the following assumptions, which are shown in Figure

The linearization

System (

Let

Double homoclinic loops

As shown in Figure

The single homoclinic loop in high-dimensional systems has been investigated by many authors (see [

A nondegenerate homoclinic orbit is called a

The rest of this paper is organized as follows. In Section

From the hypothesis

Let

By the same process as in [

Denote that

For convenience and simplicity, we use the following notations throughout Case

If (

Suppose that (

Similarly, by setting

If (

By a similar procedure, we can show that

Summing up the previous analysis, we have the following theorem.

Suppose that

If

In the region

Suppose that hypotheses

If

Let

Suppose that hypotheses

It is easy to see that

Due to

The bifurcation surface

Let

As

Still let

By a similar analysis as in Lemmas

Suppose that hypotheses

Suppose that hypotheses

Let

The bifurcation surface

As

In the following we define open regions in the neighborhood of the origin of the

Bifurcation diagram in case

Bifurcation diagram in case

Now, the previous analysis is summarized in the following three theorems, as shown in Figures

Suppose that hypotheses

has exactly one simple large 1-periodic orbit near

has a unique double large 1-periodic orbit near

has exactly two simple large 1-periodic orbits near

has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop near

does not have any large 1-periodic and large 1-homoclinic loop near

has exactly one simple large 1-periodic orbit near

has a unique double large 1-periodic orbit near

has exactly two simple large 1-periodic orbits near

has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop near

Suppose that hypotheses

has exactly one simple large 1-periodic orbit near

has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop near

has exactly two simple large 1-periodic orbits near

has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop near

has exactly one simple large 1-periodic orbit near

Suppose that hypotheses

Denote by

From the bifurcation equations (

Suppose that hypotheses

does not have any large 1-periodic orbit near

does not have any large 1-periodic orbit near

does not have any large 1-periodic orbit near

For

At last, we consider the case

For the same reason as before, we have the following theorem.

Suppose that

as

as either

Similar to Case

If (

Suppose that (

Similarly, by setting

If (

By a similar procedure, we can show that

Summarizing the previous analysis, we have the following theorem.

Suppose that

If

In the region

Suppose that hypotheses

If

Therefore, system (

Due to

By a similar analysis as in Lemma

Suppose that hypotheses

Let

In the following we define open regions in the neighborhood of the origin of the

Bifurcation diagram in case

From the previous analysis, we obtain the following three theorems, as shown in Figure

Suppose that hypotheses

does not have any large 1-periodic orbit and large 1-homoclinic loop near

has a unique large 1-homoclinic loop near

has a unique large 1-periodic orbit near

Suppose that hypotheses

has no large 1-periodic orbit and large 1-homoclinic loop near

has a unique large 1-homoclinic loop near

has a unique large 1-periodic orbit near

From the bifurcation equations, we have the following theorem easily.

Suppose that hypotheses

has no large 1-periodic orbit near

has no large 1-periodic orbit near

For

At last, we consider the case

For the same reason as before, we have the following theorem.

Suppose that

as

as

as

As

This paper is devoted to investigating the twisting bifurcations of double homoclinic loops with resonant eigenvalues in 4-dimensional systems. We give asymptotic expressions of the bifurcation surfaces and their relative positions, describe the existence regions of large 1-periodic orbits near

The research was supported by National Natural Science Foundation of China (nos. 11001041, 11101170, 11202192, 11271065 and 10926105), SRFDP (no. 200802001008), and the State Scholarship Fund of the China Scholarship Council (nos. 2011662521, and 2011842509).