The bifurcations of heteroclinic loop with one nonhyperbolic equilibrium and one hyperbolic saddle are considered, where the nonhyperbolic equilibrium is supposed to undergo a transcritical bifurcation; moreover, the heteroclinic loop has an orbit flip and an inclination flip. When the nonhyperbolic equilibrium does not undergo a transcritical bifurcation, we establish the coexistence and noncoexistence of the periodic orbits and homoclinic orbits. While the nonhyperbolic equilibrium undergoes the transcritical bifurcation, we obtain the noncoexistence of the periodic orbits and homoclinic orbits and the existence of two or three heteroclinic orbits.

In recent years, a great deal of mathematical efforts has been devoted to the bifurcation problems of homoclinic and heteroclinic orbits with high codimension, for example, the bifurcations of homoclinic or heteroclinic loop with orbit flip, the bifurcations of homoclinic or heteroclinic loop with inclination flip, and so forth; see [

Consider the following

Assuming system (

The following conditions hold in the whole paper:

It is worthy of noting that, for any integers

Let

If

The present paper is built up as follows. In Section

Let the neighborhood

From the normal form (

Consider the linear variational system

Supposing

If conditions (_{1})–(_{3}) are satisfied, then

(1) there exists a fundamental solution matrix of

(2)

Now, let

Let

Now that if

Next, we divide our establishment of the Poincaré

First, consider the map

Next, to construct the map

The final step is to compose the maps

Set

Based on the expressions of the successor functions and the implicit function theorem, we know that the equation

Firstly, we consider the case

From the above bifurcation equations, we obtain the following results immediately.

Let the conditions (_{1})–(_{3}) be true and

(i) for

(ii) there exists an

The result (i) will be proved by putting

If we assume

It follows that there exists an

This completes the proof.

There is no difficulty to see that

Assume the conditions

Theorem

If

While

Next, we only consider the case

Similarly, for

The proof is then completed.

Assume that the conditions (_{1})–(_{3}) hold and

the periodic orbit and the homoclinic loop connecting

at least one periodic orbit and the homoclinic loop connecting

a unique periodic orbit and the homoclinic loop connecting

By Theorem

(i) Let

For

Now we turn to the case

(ii) For

Choosing

(iii)

This completes the proof.

Now, we turn to discussing the bifurcations of the heteroclinic loop for

Firstly, we take into account the case

With similar arguments to

Suppose the conditions (_{1})–(_{3}) hold,

(i) if

(ii) there exists an

Suppose hypotheses (_{1})–(_{3}) hold,

It is easy to see that homoclinic loop connecting

Finally, we consider the case

Assume the conditions (_{1})–(_{3}) are true,

there exists a surface

there exists a region in the

(i) If

(ii) If

All the heteroclinic orbits joining

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper is supported by National Natural Science Foundation of China (nos