Bifurcations and Stability of Nondegenerated Homoclinic Loops for Higher Dimensional Systems

By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence of k-homoclinic loop and k-periodic orbit is obtained. Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.


Introduction
With the rapid development of nonlinear science, in the studies of many fields of research and application of medicine, life sciences and many other disciplines, there are a lot of variety high-dimensional nonlinear dynamical systems with complex dynamic behaviors. Homoclinic and heteroclinic orbits and the corresponding bifurcation phenomenons are the most important sources of complex dynamic behaviors, which occupy a very important position in the research of high-dimensional nonlinear systems. We know that in the study of high-dimensional dynamical systems of infectious diseases and population ecology we tend to ignore the stability switches and chaos when considered much more the nonlinear incidence rate, population momentum, strong nonlinear incidence rate, and so forth. The existence of transversal homoclinic orbits implies that chaos phenomenon occur; therefore, it is of very important significance to study the cross-sectional of homoclinic orbits and the preservation of homoclinic orbits for the system in small perturbation.
In addition, in the study of infectious diseases and population ecology systems, we sometimes require the existence of periodic orbits. And, homoclinic and heteroclinic orbits bifurcate to periodic orbits in a small perturbation means that we can get the required periodic solution only by adding a small perturbation when using the similar system which exists homoclinic or heteroclinic orbits to represent the natural system. This also explains the importance of homoclinic and heteroclinic orbits bifurcating periodic orbits in realworld applications.
Therefore, by using the research methods and theoretical results of qualitative and bifurcation problems of highdimensional systems, especially the results of homoclinic and heteroclinic orbits and their bifurcations for the systems, to study the high-dimensional infectious disease dynamics and population ecology systems to reveal the complex dynamical behavior of the nonlinear dynamical systems and the corresponding reality systems is essential.
About the study of bifurcation problems of homoclinic and heteroclinic loops for two-dimensional systems, a large number of papers were obtained and achieved many good results (for some results see [1][2][3][4][5][6]); but for higherdimensional nonlinear systems, due to the complexity, the results we see today are not abundant. Chow S. N., Deng B., and Fiedler B. discussed the bifurcation of non-degenerated homoclinic loop [7], but the research method is abstract, and the results are focused on the theory. Some subsequent studies are mostly based on the traditional Poincaré map construction method. Zhu [8] discussed the non-degenerated bifurcation problems of homoclinic loop of the systeṁ= ( , ) + ( , , ). Compared with [7], paper [8] described the bifurcation surface and bifurcation phenomenon by using the inherent eigenvalue, so that the results possess good operability.
In this paper, the bifurcation and stability problems of non-degenerated homoclinic loops under non-resonant and resonant conditions for high dimensional systeṁ= ( ) + ( , ) are considered. The method to establish the local coordinates system in the tubular neighborhood of homoclinic loop used in [8] is simplified here. In [8], the author used the generalized Floquet method to establish local coordinate systems and Poincaré map. Here, we use the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local coordinates system of the perturbed system in the small neighborhood of the homoclinic orbit. The Poincaré Maps and bifurcation equations obtained by this method are more simple and convenient for analysis than [8]. Besides, this method does not only have important significance in theory, but it can also be operated well in applications.

Hypotheses
Suppose the following system: where ≥ 4, ∈ R + satisfies the following hypotheses.

Local Coordinates
Suppose that the neighborhood of = 0 is small enough and (H1)∼(H3) are established, then, for | | is small enough, we can introduce a change such that system (3) has the following form in : ∈ R −1 , V ∈ R −1 , and * means transposition. Moreover, in , we suppose that the instable manifold, the stable manifold, the strong instable manifold, the strong stable manifold and the local homoclinic orbits have the following forms, respectively. Consider the linear systeṁ = ( ( ( ))) .

Poincaré Maps and Bifurcation Equations
: | |, | |, | |, |V| < 2 } be the cross sections of Γ at = and = − , respectively, where, is small enough such that 0 , 1 ⊂ Figure 1. Figure 1 solution of (3) in the small neighborhood of = 0 (similarly, we can set up the map by the flow of the linear system of (4) in the small neighborhood of = 0), 1 : 1 → 0 is set up by the solution manifold of (3) in the tubular neighborhood of Γ.

Set Up
4
If > 0, then the solution (22) corresponds to a 1periodic orbit of the system (3), that is, the homoclinic orbit Γ bifurcates to a periodic orbit.
The proof is complete. The proof is complete. Now, we consider the nonexistence of -homoclinic loop and -periodic orbit, where > 1. Firstly, we consider the case of = 2.
And, so, for any > 1, we have the following.

Resonance Bifurcation
We say that the homoclinic loop is Resonance if 1 = 1 . For convenience, we assume the resonant condition has the following form.
(iv) The system (3) has exactly a 1-homoclinic orbit and a 1-periodic orbit near Γ if and only if ∈ Σ 2 .
Similarly, we can define the corresponding 1 ( ) and 2 ( ), and the corresponding Σ 1 and Σ 2 , to obtain the following theorem.  Now, we consider the nonexistence of -homoclinic loop and -periodic orbit, where > 1. We may assume that = 2.
If (32) has the solution 1 > 0, 2 > 0, then (32) turns to Let ( 2 ) and ( 2 ) be the left and right of the above formula, respectively, then ( 2 ) and ( 2 ) are tangent at some point if and only if (36) and the following formulas are fulfilled.

Stability
Now, we consider the stability of homoclinic loop Γ.
Besides, by the above discussion, we can get the following.
Proof. In fact, from [1][2][3], there always exist a transformation coordinates, such that the system (3) has the following form in a small neighborhood of the origin: Remark 12. Because the divergence integration is the invariant under the 2 transformation (refer [4,16]), so the function (⋅) and ( ) of the divergence integration can be thought of the original forms of (3). , therefore, − 12 21 = 1, and by 21 = −1, we get 12 = 1/ .
The proof is complete.
Combined with the Theorem 9, we have the following.