Bifurcations of a Homoclinic Orbit to Saddle-Center in Reversible Systems

and Applied Analysis 3 the unstable manifold in U0 by using the method introduced in Zhu 9 . According to the invariance and symmetry of these manifolds, we can deduce that system 2.1 has the following form inU0: ẋ x ( λ ( , μ ) O 1 ) , ẏ y −λ , μ O 1 , u̇ v ( ω ( , μ ) O 1 ) u ( O x O ( y ) O u ) O ( xy ) , v̇ u −ω , μ O 1 ) − vO x Oy O v ) −Oxy, 3.1 where λ 0, μ λ, ω 0, μ ω,O 1 O x O y O u O v , the system is Cr−2, and the corresponding involution acts as R x, y, u, v y, x, v, u . In fact, by a linear transformation, system 2.1 takes the form in a small neighborhood ofU0 as follows: ẋ λ ( , μ ) x O 2 , ẏ −λ , μy O 2 , u̇ ω ( , μ ) v O 2 , v̇ −ω , μu O 2 , 3.2 and R x, y, u, v y, x, v, u . By the invariant manifold theorem, we know that there exist a local C center-stable manifold W ,μ {z x, y, u, v : x x ,μ y, u, v , x ,μ 0, 0, 0 0, Dx ,μ 0, 0, 0 0, z ∈ U0}, a local C center-unstable manifold W ,μ {z x, y, u, v : y y ,μ x, u, v , y cu ,μ 0, 0, 0 0, Dy cu ,μ 0, 0, 0 0, z ∈ U0} and RW ,μ W ,μ. By the straightening coordinate transformation which is similar to that of 1, 6, 9 , now we straighten the local manifolds W ,μ and W cu ,μ, such that W cs ,μ {z ∈ U0 : x 0}, W ,μ {z ∈ U0 : y 0}. Notice that the invariance of W ,μ and W cu ,μ implies the local invariance of {z ∈ U0 : x 0} and {z ∈ U0 : y 0}, respectively, which produces that, inU0, ẋ x ( λ ( , μ ) O 1 ) , ẏ y −λ , μ O 1 . 3.3 Now the system is Cr−1 and still reversible. By using a similar procedure to straighten the local Cr−1 stable manifold W ,μ and unstable manifold W u ,μ, and the invariance and symmetry of these two local manifolds that means the transformation is also symmetric , we get system 3.1 . Clearly, corresponding to system 3.1 , the center manifoldW ,μ is locally in the u-v plane, and the stable manifold W ,μ resp., unstable manifold W u ,μ is locally the y-axis resp., x-axis when they are confined in U0. Define r −T δ, 0, 0, 0 ∗, r T 0, δ, 0, 0 ∗ for T 1, where δ > 0 is small enough such that { x, y, u, v ∗ : |x|, |y|, |u2 v2|1/2 < 2δ} ⊂ U0. 4 Abstract and Applied Analysis Consider the linear variational system


Introduction
In recent years, there are much interest in the phenomenon of homoclinics and heteroclinics in reversible dynamical systems because of their extensive applications in mechanics, fluids, and optics 1-10 .For example, Klaus and Knobloch 4 considered a homoclinic orbit to saddle-center with two-parameter families of non-Hamiltonian reversible vector fields by Lin's method.They derived the occurrence of 1-homoclinic orbits to the center manifold.Liu et al. 6 studied a singular perturbation system with action-angle variable and the unperturbed system was assumed to possess a saddle-center equilibrium in a general system without reversible or Hamiltonian structure.Mielke et al. 7 investigated bifurcations of homoclinic orbit to saddle-center in 4-dimensional reversible Hamiltonian systems.By using the Poincaré map and a special normal form, they detected the existence of Nhomoclinic orbits to the equilibrium, N-periodic orbits, and chaotic behavior near the primary homoclinic orbit.As for purely Hamiltonian system in R 4 , similar results are known from Koltsova and Lerman 2, 5 .In all of these papers the underlying Hamiltonian structure was heavily considered.Especially it was used to detect multiround orbits.Note that, the Hamiltonian can make the dynamics constrict in three-dimensional manifold in a zero level set.In this paper, we use the method originated by Zhu 9 , by constructing a local moving coordinate system and Poincaré map near the primary homoclinic orbit, the existence of transversal homoclinic orbits and periodic orbits bifurcated from the primary homoclinic orbit are obtained in a 4-dimensional reversible system.It is worth to mention that a new kind of moving coordinates is introduced firstly in our paper in order to simplify and facilitate the reversible system.
The remainder of this paper is organized as follows.Section 2 contains the assumptions for the perturbed and unperturbed system.The local coordinate moving frame, cross sections, and Poincaré map are set up in Section 3. Finally, we obtain the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding surfaces with different condimensions in Sections 4 and 5.

The General Setup
Consider the following system and the corresponding unperturbed system ż f z , 2.2 where 0 < 1, l ≥ 1, f : R 4 → R 4 and g : R 4 × R → R 4 are C r r ≥ 4 , f 0 g 0; μ 0 and μ is a parameter.Also, we need the following assumptions.
A1 System 2.1 is reversible with respect to the linear involution R such that dim Fix R 2, f Rz Rf z g Rz; μ Rg z; μ 0 for all z ∈ R 4 and μ ∈ R l .Note that, throughout the paper, we will denote R-symmetric orbit as symmetric orbit.A2 The origin O is a saddle-center equilibrium of 2.

Local Moving Frame and Poincar é Map
Suppose the neighborhood U 0 of O is small enough, we can firstly straighten the centerstable manifold, the center-unstable manifold, subsequently, then the stable manifold and the unstable manifold in U 0 by using the method introduced in Zhu 9 .According to the invariance and symmetry of these manifolds, we can deduce that system 2.1 has the following form in U 0 : , and the corresponding involution acts as R x, y, u, v y, x, v, u .In fact, by a linear transformation, system 2.1 takes the form in a small neighborhood of U 0 as follows: and R x, y, u, v y, x, v, u .By the invariant manifold theorem, we know that there exist a local C r center-stable manifold W cs ,μ {z x, y, u, v : x x cs ,μ y, u, v , x cs ,μ 0, 0, 0 0, Dx cs ,μ 0, 0, 0 0, z ∈ U 0 }, a local C r center-unstable manifold W cu ,μ {z x, y, u, v : y y cu ,μ x, u, v , y cu ,μ 0, 0, 0 0, Dy cu ,μ 0, 0, 0 0, z ∈ U 0 } and RW cs ,μ W cu ,μ .By the straightening coordinate transformation which is similar to that of 1, 6, 9 , now we straighten the local manifolds W cs ,μ and Notice that the invariance of W cs ,μ and W cu ,μ implies the local invariance of {z ∈ U 0 : x 0} and {z ∈ U 0 : y 0}, respectively, which produces that, in U 0 ,

3.3
Now the system is C r−1 and still reversible.By using a similar procedure to straighten the local C r−1 stable manifold W s ,μ and unstable manifold W u ,μ , and the invariance and symmetry of these two local manifolds that means the transformation is also symmetric , we get system 3.1 .Clearly, corresponding to system 3.1 , the center manifold W c ,μ is locally in the u-v plane, and the stable manifold W s ,μ resp., unstable manifold W u ,μ is locally the y-axis resp., x-axis when they are confined in U 0 .

3.5
Based on the invariance and symmetry of manifolds W cs and W cu , it is easy to know that system 3.4 has a fundamental solution matrix Z t Actually, in the resulting coordinates, W s ∩ U 0 and W u ∩ U 0 are y-axis and x-axis, respectively, combining with the symmetry, it follows that On the other hand, in a small tubular neighborhood of the homoclinic loop Γ, the center-unstable manifold W cu resp., center-stable manifold W cs can be foliated into a family of leaves, each is a 2-dimensional surface and asymptotic to W c as the base point z r t ∈ Γ tends to O as t → −∞ resp., ∞ .Notice that the limit of the linearization 3.
1, 0 see Figure 1 for details .Thus, if we take solutions z 3 t and z 4 t in T r t W cu satisfying z 3 −T 0, 0, 1, 0 * , z 4 −T 0, 0, 0, 1 * , then, restricted to the u-v plane, z 3 −T is the unit tangent direction of the closed orbit Γ at u, v 0, a 0, ω −1 .In addition, the restriction of z 4 −T is its unit exterior normal direction at the same point.By the reversibility, it is easy to obtain z 3 T 0, 0, 0, −1 * , z 4 T 0, 0, 1, 0 * .Finally, if we choose a solution z 1 t ∈ T r t W cu c with z 1 −T 0, 1, 0, 0 * , then the symmetry says that z 1 T 1, 0, 0, 0 * .Therefore, we have demonstrated the existence of the fundamental matrix Z t with the specified properties.
Remark 3.1.In the following, we will regard z 1 t , z 2 t , z 3 t , z 4 t as a moving coordinate in a small tubular neighborhood of Γ.This new kind of moving frame is firstly introduced for the homoclinic orbit to a saddle-center, which is the extension of the corresponding coordinates built in 1, 6, 9 for the homoclinic orbit to a saddle.The explicit advantage is that, these coordinate vectors inherit and exhibit the geometrical and dynamical properties of those invariant manifolds.As mentioned above, they will greatly simplify the original reversible system.
δ/2} ⊂ U 0 be the cross sections of Γ at t −T and t T , respectively.Now we turn to seek the new coordinates of q 0 ∈ Σ u and q 1 ∈ Σ s see Figure 2 for details under the transformation z t r t Z t • N, where N n 1 , 0, n 3 , n 4 * .Take which are solved by Putting the transformation z t r t Z t • N into 2.1 , we get

3.12
That implies Z t • Ṅ g r t , μ O N 2 , which is C r−3 .Then multiplying the resulting equation by 4 * and integrating it on both sides from −T to T , we have the regular map Π R : Σ u → Σ s : where M i T, μ T −T φ * i g i r t , μ dt are the Melnikov functions.For conciseness, we will denote λ , μ λ, ω , μ ω.Now we consider the local map Π S induced by the flow 3.1 in U 0 , where

3.14
Let τ be the flying time from q 1 to q 0 , by variation of constants formula, we can get the following expression:
Denote s e −λτ , then we get Π S : Σ s → Σ u defined by

3.16
Combining the maps 3.13 and 3.16 , we obtain the Poincaré map Π Π R • Π S : Σ s → Σ s defined as 3.17

Existence of 1-Homoclinic Orbit and 1-Periodic Orbit
Let G q 1 Π q 1 − q 1 be the displacement function.Based on 3.11 , 3.17 , and the new coordinate of q 1 n 1 1 , 0, n 1 3 , n 1  4 , we see the small zero point s, u 1 , v 1 of G will satisfy the following equations:

4.1
Due to the coordinate transformations introduced in U 0 at the beginning of Section 3, the unstable manifold and the stable manifold are locally x-axis and y-axis, respectively, so it is evident that, near Γ, system 2.1 has a symmetric 1-homoclinic orbit to O if and only if 4.1 have a solution s, u 1 , v 1 with s u 1 v 1 0, and system 2.1 has an 1-homoclinic orbit to a periodic orbit on the center manifold and an 1-periodic orbit if and only if 4.1 have a solution s, u 1 , v 1 with s 0, 1, and C 3 in , μ , and has a solution s u 1 v 1 0 as 0, thus we can rescale s 2 s, u 1 2 u, v 1 2 v, such that system 4.1 is reformulated as where O depends on , s, u, v and μ.
Now the following results are verified directly by the implicit function theorem.

Existence of Symmetric 1-Periodic Orbit
In this section, we turn to seek the existence of symmetric 1-periodic orbit.Note that, a 1periodic orbit L near Γ is symmetric if and only Due to 3.15 , it is equivalent to
Remark 5.2.Similarly, if we take s e −λτ u 1 v 1 0 in 3.15 , then x 1 y 0 u 0 v 0 u 1 v 1 0 i.e., condition 5.1 , it means that there is a symmetric 1-homoclinic orbit to O if and only if s u 1 v 1 0 in Section 4.
Remark 5.3.In Theorem 4.2 resp., Theorem 5.1 , the geometric meaning of k is that, confined in U 0 , the u-v component of the above 1-periodic orbit bifurcated from Γ makes circle k resp., 2k times around the saddle-center.

Figure 1 :
Figure 1: The geometry of vectors on the center manifold.

Figure 2 :
Figure 2: Setup of cross sections and Poincaré map.

Remark 5 . 4 .
From x 1 / √ s δ √ s 1 O δ , s O 2 and the constitution of ρ 11 and ρ 21 , it is easy to know that the necessary condition for ρ 11 ρ 21 is O 1 y 1 O 2 y 1 .
Note that, O is a symmetric equilibrium, and the eigenvalues of the Jacobian matrix A are symmetric with respect to the imaginary axis.