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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.

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 [

Recently, Zhang et al. in [

As mentioned in [

We first consider a

(H1)

(H2)

(H3)

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

This section treats mainly the establishment of Poincaré return map with two steps. To begin we first need to transform system (

It is well known that there are always two

Owing to the above straightness of the invariant manifolds, it is easy to find some moment

Transition maps.

In the following part we construct the map

There exists a fundamental solution matrix

Notice that the tangent subspace

As to

The matrix

In fact

Equation (

In order to combine

With all of the above, the Poincaré return map is given as

From the definition of the

From

Put

Suppose that

Clearly

In the following part we restrict our attention on the case

In order to well solve (

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

Suppose that

It is clear that

Suppose that

From Lemma

Suppose that

Rank

Double 1-periodic orbit

Two 1-periodic orbits

No 1-periodic orbit

We know that the existence of a double 1-periodic orbit corresponds to the equations

From the above proof, we see that, when

Assume that the hypotheses of Theorem

As

Suppose

For

For

system (

For

For

For

When

The result of the cases

If

In case of

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

Notice that, in Theorem

1-homoclinic orbit (1-H) and (1-OH).

There still exist some double 1-periodic orbits or triple 1-periodic orbits for the case

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