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

Abstract

The bifurcations near a primary homoclinic orbit to a saddle-center are investigated in a 4-dimensional reversible system. By establishing a new kind of local moving frame along the primary homoclinic orbit and using the Melnikov functions, the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding codimension 1 or codimension 3 surfaces, are obtained.

1. 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 𝑁-homoclinic orbits to the equilibrium, 𝑁-periodic orbits, and chaotic behavior near the primary homoclinic orbit. As for purely Hamiltonian system in ℝ⁴, 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.

2. The General Setup

Consider the following system 𝑧̇=𝑓(𝑧)+𝜖𝑔𝑧;𝜇,𝑧∈ℝ⁴,𝜇∈ℝ^𝑙 (2.1) and the corresponding unperturbed system 𝑧̇=𝑓(𝑧), (2.2) where 0<𝜖≪1, 𝑙≥1, 𝑓∶ℝ⁴→ℝ⁴ and 𝑔∶ℝ⁴×ℝ→ℝ⁴ 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(Fix𝑅)=2, 𝑓(𝑅𝑧)+𝑅𝑓(𝑧)=𝑔(𝑅𝑧;𝜇)+𝑅𝑔(𝑧;𝜇)=0 for all 𝑧∈ℝ⁴ 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 (2.2), 𝑂∈Fix𝑅. More precisely, the Jacobian matrix 𝐴≜𝐷𝑓(0) has a pair of purely imaginary eigenvalues ±𝑖𝜔 and two nonzero real eigenvalues, that is, 𝜎(𝐴)={±𝑖𝜔,±𝜆} with 𝜔,𝜆>0.

(A3) System (2.2) has a symmetric homoclinic orbit Γ={𝑧=𝑟(𝑡)∶𝑡∈ℝ}, where 𝑟(±∞)=𝑂.

Note that, 𝑂 is a symmetric equilibrium, and the eigenvalues of the Jacobian matrix 𝐴 are symmetric with respect to the imaginary axis. Thus, assumption (A2) describes a scenario that is structurally stable. Furthermore, the saddle-center 𝑂 has one-dimensional stable manifold 𝑊^𝑠_𝜖,𝜇(𝑂) and one-dimensional unstable manifolds 𝑊^𝑢_𝜖,𝜇(𝑂) (abbr. 𝑊^𝑠 and 𝑊^𝑢 as 𝜖=0), and a two-dimensional center manifold 𝑊^𝑐_𝜖,𝜇(𝑂) (abbr. 𝑊^𝑐 as 𝜖=0) for 𝜖 close to 0. All of them are 𝐶^𝑟. Confined to 𝑊^𝑐_𝜖,𝜇(𝑂), 𝑂 is a center. The reversibility implies that 𝑊^𝑢_𝜖,𝜇(𝑂)=𝑅𝑊^𝑠_𝜖,𝜇(𝑂) and 𝑊^𝑐_𝜖,𝜇(𝑂)=𝑅𝑊^𝑐_𝜖,𝜇(𝑂), and hence, the homoclinic orbit Γ is symmetric, that is, 𝑅Γ=Γ.

3. Local Moving Frame and Poincaré Map

Suppose the neighborhood 𝑈₀ of 𝑂 is small enough, we can firstly straighten the center-stable manifold, the center-unstable manifold, subsequently, then the stable manifold and 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 in 𝑈₀: ̇𝜆(1),̇(1),̇𝜔(1)𝑂(𝑥)𝑦(𝑢),̇(1)𝑂(𝑥)𝑦(𝑣),𝑥=𝑥𝜖,𝜇+𝑂𝑦=𝑦−𝜆𝜖,𝜇+𝑂𝑢=𝑣𝜖,𝜇+𝑂+𝑢+𝑂+𝑂+𝑂𝑥𝑦𝑣=𝑢−𝜔𝜖,𝜇+𝑂−𝑣+𝑂+𝑂−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 𝐶^𝑟−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 of 𝑈₀ as follows: ̇(2),̇(2),̇(2),̇(2),𝑥=𝜆𝜖,𝜇𝑥+𝑂𝑦=−𝜆𝜖,𝜇𝑦+𝑂𝑢=𝜔𝜖,𝜇𝑣+𝑂𝑣=−𝜔𝜖,𝜇𝑢+𝑂 (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^𝑟 center-stable manifold 𝑊^𝑐𝑠_𝜖,𝜇={𝑧=(𝑥,𝑦,𝑢,𝑣)∶𝑥=𝑥^𝑐𝑠_𝜖,𝜇(𝑦,𝑢,𝑣),𝑥^𝑐𝑠_𝜖,𝜇(0,0,0)=0,𝐷𝑥^𝑐𝑠_𝜖,𝜇(0,0,0)=0,𝑧∈𝑈₀}, a local 𝐶^𝑟 center-unstable manifold 𝑊^𝑐𝑢_𝜖,𝜇={𝑧=(𝑥,𝑦,𝑢,𝑣)∶𝑦=𝑦^𝑐𝑢_𝜖,𝜇(𝑥,𝑢,𝑣),𝑦^𝑐𝑢_𝜖,𝜇(0,0,0)=0,𝐷𝑦^𝑐𝑢_𝜖,𝜇(0,0,0)=0,𝑧∈𝑈₀} and 𝑅𝑊^𝑐𝑠_𝜖,𝜇=𝑊^𝑐𝑢_𝜖,𝜇.

By the straightening coordinate transformation which is similar to that of [1, 6, 9], now we straighten the local manifolds 𝑊^𝑐𝑠_𝜖,𝜇 and 𝑊^𝑐𝑢_𝜖,𝜇, such that 𝑊^𝑐𝑠_𝜖,𝜇={𝑧∈𝑈₀∶𝑥=0},𝑊^𝑐𝑢_𝜖,𝜇={𝑧∈𝑈₀∶𝑦=0}.

Notice that the invariance of 𝑊^𝑐𝑠_𝜖,𝜇 and 𝑊^𝑐𝑢_𝜖,𝜇 implies the local invariance of {𝑧∈𝑈₀∶𝑥=0} and {𝑧∈𝑈₀∶𝑦=0}, respectively, which produces that, in 𝑈₀, ̇𝜆(1),̇(1).𝑥=𝑥𝜖,𝜇+𝑂𝑦=𝑦−𝜆𝜖,𝜇+𝑂 (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_𝜖,𝜇 and unstable manifold 𝑊^𝑢_𝜖,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_𝜖,𝜇 is locally in the 𝑢-𝑣 plane, and the stable manifold 𝑊^𝑠_𝜖,𝜇 (resp., unstable manifold 𝑊^𝑢_𝜖,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.

Consider the linear variational system 𝑧̇=𝐷𝑓(𝑟(𝑡))𝑧, (3.4) and its adjoint system 𝜙̇=−𝐷𝑓(𝑟(𝑡))^∗𝜙. (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)=(𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡),𝑧₄(𝑡)) satisfying 𝑧₁(𝑡)∈(𝑇_𝑟(𝑡)𝑊^𝑐𝑢)^𝑐, 𝑧₂(𝑡)=−𝑟(̇𝑡)/|𝑟(̇𝑇)|∈𝑇_𝑟(𝑡)(𝑊^𝑠∩𝑊^𝑢), 𝑧₃(𝑡),𝑧₄(𝑡)∈𝑇_𝑟(𝑡)𝑊^𝑐𝑢 and 𝑍(−𝑇)=(𝑇)=0−100100000100001,𝑍10000100000100−10. (3.6)

Actually, in the resulting coordinates, 𝑊^𝑠∩𝑈₀ and 𝑊^𝑢∩𝑈₀ are 𝑦-axis and 𝑥-axis, respectively, combining with the symmetry, it follows that 𝑧₂(𝑇)=(0,1,0,0)^∗,𝑧₂(−𝑇)=(−1,0,0,0)^∗. (3.7)

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 (3.4) of system (2.2) with respect to Γ as 𝑡→±∞ is 𝑥̇=𝜆𝑥,𝑦̇=−𝜆𝑦,𝑢̇=𝜔𝑣,𝑣̇=−𝜔𝑢. (3.8) When confined on 𝑊^𝑐, it becomes the following subsystem 𝑢̇=𝜔𝑣,𝑣̇=−𝜔𝑢, (3.9) which is reversible with the involution (𝑢,𝑣)→(𝑣,𝑢). Obviously, for any 𝑎>0 and 𝑡₀∈𝑅, 𝑢=𝑢(𝑡)=𝑎sin𝜔(𝑡−𝑡₀), 𝑣=𝑣(𝑡)=𝑎cos𝜔(𝑡−𝑡₀) is a solution of (3.9), which defines a closed orbit Γ on 𝑊^𝑐. More precisely, its tangent vectors 𝑢₁(𝑡)=𝑢̇(𝑡)=𝑎𝜔cos𝜔(𝑡−𝑡₀), 𝑣₁(𝑡)=𝑣(̇𝑡)=−𝑎𝜔sin𝜔(𝑡−𝑡₀) and its normal vectors 𝑢₂(𝑡)=−𝑣(̇𝑡)=𝑎𝜔sin𝜔(𝑡−𝑡₀), 𝑣₂(𝑡)=𝑢̇(𝑡)=𝑎𝜔cos𝜔(𝑡−𝑡₀) are the solutions of (3.9). Choose some appropriate 𝑎 and 𝑡₀ such that 𝑎𝜔=1,𝜔(−𝑇−𝑡₀)=2𝑘𝜋 for some 𝑘∈𝑍, then (𝑢(−𝑇),𝑣(−𝑇))=(0,𝑎)=(0,𝑤⁻¹),𝑧₃(−𝑇)=(𝑢₁(−𝑇),𝑣₁(−𝑇))=(1,0),𝑧₄(−𝑇)=(𝑢₂(−𝑇),𝑣₂(−𝑇))=(0,1). Based on the reversibility, we have (𝑢(𝑇),𝑣(𝑇))=(𝑎,0)=(𝑤⁻¹,0),𝑧₃(𝑇)=(𝑢₁(𝑇),𝑣₁(𝑇))=(0,−1),𝑧₄(𝑇)=(𝑢₂(𝑇),𝑣₂(𝑇))=(1,0) (see Figure 1 for details). Thus, if we take solutions 𝑧₃(𝑡) and 𝑧₄(𝑡) in 𝑇_𝑟(𝑡)𝑊^𝑐𝑢 satisfying 𝑧₃(−𝑇)=(0,0,1,0)^∗,𝑧₄(−𝑇)=(0,0,0,1)^∗, then, restricted to the 𝑢-𝑣 plane, 𝑧₃(−𝑇) is the unit tangent direction of the closed orbit Γ at (𝑢,𝑣)=(0,𝑎)=(0,𝜔⁻¹). In addition, the restriction of 𝑧₄(−𝑇) is its unit exterior normal direction at the same point. By the reversibility, it is easy to obtain 𝑧₃(𝑇)=(0,0,0,−1)^∗, 𝑧₄(𝑇)=(0,0,1,0)^∗.

Figure 1 

The geometry of vectors on the center manifold.

Finally, if we choose a solution 𝑧₁(𝑡)∈(𝑇_𝑟(𝑡)𝑊^𝑐𝑢)^𝑐 with 𝑧₁(−𝑇)=(0,1,0,0)^∗, then the symmetry says that 𝑧₁(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 (𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡),𝑧₄(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.

Let Σ^𝑢={(𝑥,𝑦,𝑢,𝑣)∣𝑥=𝛿,|𝑦|,|𝑢²+𝑣²|^1/2≤𝛿/2}⊂𝑈₀ and Σ^𝑠={(𝑥,𝑦,𝑢,𝑣)∣𝑦=𝛿,|𝑥|,|𝑢²+𝑣²|^1/2≤𝛿/2}⊂𝑈₀ be the cross sections of Γ at 𝑡=−𝑇 and 𝑡=𝑇, respectively.

Now we turn to seek the new coordinates of 𝑞₀∈Σ^𝑢 and 𝑞₁∈Σ^𝑠 (see Figure 2 for details) under the transformation 𝑧(𝑡)=𝑟(𝑡)+𝑍(𝑡)⋅𝑁, where 𝑁=(𝑛₁,0,𝑛₃,𝑛₄)^∗. Take 𝑞₀≜𝑥₀,𝑦₀,𝑢₀,𝑣₀=𝑟(−𝑇)+𝑍(−𝑇)⋅𝑛⁰₁,0,𝑛⁰₃,𝑛⁰₄^∗,𝑞₁≜𝑥₁,𝑦₁,𝑢₁,𝑣₁=𝑟(𝑇)+𝑍(𝑇)⋅𝑛¹₁,0,𝑛¹₃,𝑛¹₄^∗, (3.10) which are solved by 𝑥₀=𝛿,𝑛⁰₁=𝑦₀,𝑛⁰₃=𝑢₀,𝑛⁰₄=𝑣₀;𝑦₁=𝛿,𝑛¹₁=𝑥₁,𝑛¹₃=−𝑣₁,𝑛¹₄=𝑢₁. (3.11)

Figure 2 

Setup of cross sections and Poincaré map.

Putting the transformation 𝑧(𝑡)=𝑟(𝑡)+𝑍(𝑡)⋅𝑁 into (2.1), we get 𝑟̇(𝑡)+𝑍̇(𝑡)(𝑡)⋅̇(𝑟(𝑡)(𝑡))𝑟(𝑡)(𝑡)(𝑟(𝑡(𝑟(𝑡𝑍(𝑡)𝑟(𝑡))⋅𝑁+𝑍𝑁=𝑓+𝑍⋅𝑁+𝜖𝑔+𝑍⋅𝑁,𝜇=𝑓))+𝐷𝑓))⋅𝑁+𝜖𝑔,𝜇+𝑂(𝜖+𝑁². (3.12) That implies 𝑍(𝑡)⋅𝑁̇=𝜖𝑔(𝑟(𝑡),𝜇)+𝑂((𝜖+𝑁)²), which is 𝐶^𝑟−3. Then multiplying the resulting equation by 𝑍⁻¹(𝑡)=(𝜙^∗₁,𝜙^∗₂,𝜙^∗₃,𝜙^∗₄)^∗ and integrating it on both sides from −𝑇 to 𝑇, we have the regular map Π_𝑅∶Σ^𝑢→Σ^𝑠: 𝑛_𝑖(𝑇)=𝑛_𝑖(−𝑇)+𝜖𝑀_𝑖𝑇,𝜇+𝑂(𝜖+𝑁(−𝑇))²,𝑖=1,3,4, (3.13) where 𝑀_𝑖(𝑇,𝜇)=∫^𝑇_−𝑇𝜙^∗_𝑖𝑔_𝑖(𝑟(𝑡),𝜇)𝑑𝑡 are the Melnikov functions.

For conciseness, we will denote 𝜆(𝜖,𝜇)≜𝜆,𝜔(𝜖,𝜇)≜𝜔. Now we consider the local map Π_𝑆 induced by the flow (3.1) in 𝑈₀, where Π_𝑆∶Σ^𝑠⟶Σ^𝑢,𝑞₁=𝑛¹₁,0,𝑛¹₃,𝑛¹₄⟶𝑞₀=𝑛⁰₁,0,𝑛⁰₃,𝑛⁰₄. (3.14) Let 𝜏 be the flying time from 𝑞₁ to 𝑞₀, by variation of constants formula, we can get the following expression: 𝑥₁=𝑥(𝑇)=𝑥₀𝑒^−𝜆𝜏1+𝜌₁,𝑦₀=𝑦(𝑇+𝜏)=𝑦₁𝑒^−𝜆𝜏1+𝜌₂,𝑢₀=𝑢(𝑇+𝜏)=𝑢₁cos(𝜔𝜏)+𝑣₁sin(𝜔𝜏)1+𝜌₃+𝜌₅,𝑣₀=𝑣(𝑇+𝜏)=−𝑢₁sin(𝜔𝜏)+𝑣₁cos(𝜔𝜏)1+𝜌₄+𝜌₆, (3.15) where 𝜌_𝑖=𝜌_𝑖(𝑥₀,𝑦₁,𝑢₁,𝑣₁,𝜏,𝜖,𝜇), 𝜌_𝑖∈𝐶⁰, 𝑖=1,…,6, and 𝜌_𝑖=𝑂(𝛿) for 𝑖=1,2,3,4, 𝜌_𝑗=𝑂(𝛿²𝑒^−𝜆𝜏) for 𝑗=5,6.

Denote 𝑠=𝑒^−𝜆𝜏, then we get Π_𝑆∶Σ^𝑠→Σ^𝑢 defined by 𝑛⁰₁=𝛿𝑠1+𝜌₂,𝑛⁰₃=𝑛¹₄𝜔cos𝜆log𝑠+𝑛¹₃𝜔sin𝜆log𝑠1+𝜌₃+𝜌₅,𝑛⁰₄=𝑛¹₄𝜔sin𝜆log𝑠−𝑛¹₃𝜔cos𝜆log𝑠1+𝜌₄+𝜌₆. (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 𝑛₁(𝑇)=𝛿𝑠1+𝜌₂+𝜖𝑀₁𝑛𝑇,𝜇+ℎ.𝑜.𝑡.,₃(𝑇)=𝑛¹₄𝜔cos𝜆log𝑠+𝑛¹₃𝜔sin𝜆log𝑠1+𝜌₃+𝜖𝑀₃𝑇,𝜇+𝜌₅𝑛+ℎ.𝑜.𝑡.,₄(𝑇)=𝑛¹₄𝜔sin𝜆log𝑠−𝑛¹₃𝜔cos𝜆log𝑠1+𝜌₄+𝜖𝑀₄𝑇,𝜇+𝜌₆+ℎ.𝑜.𝑡. (3.17)

4. Existence of 1-Homoclinic Orbit and 1-Periodic Orbit

Let 𝐺(𝑞₁)≜Π(𝑞₁)−𝑞₁ be the displacement function. Based on (3.11), (3.17), and the new coordinate of 𝑞₁=(𝑛¹₁,0,𝑛¹₃,𝑛¹₄), we see the small zero point (𝑠,𝑢₁,𝑣₁) of 𝐺 will satisfy the following equations: 𝛿𝑠𝜌₂−𝜌₁+𝜖𝑀₁𝑇,𝜇+𝑂|𝑠,𝜖|²𝑣=0,₁+𝑢₁𝜔cos𝜆log𝑠−𝑣₁𝜔sin𝜆log𝑠1+𝜌₃+𝜖𝑀₃𝑇,𝜇+𝜌₅𝑢+ℎ.𝑜.𝑡.=0,₁−𝑢₁𝜔sin𝜆log𝑠+𝑣₁𝜔cos𝜆log𝑠1+𝜌₄−𝜖𝑀₄𝑇,𝜇−𝜌₆+ℎ.𝑜.𝑡.=0. (4.1)

Due to the coordinate transformations introduced in 𝑈₀ 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,𝑢₁,𝑣₁) with 𝑠=𝑢₁=𝑣₁=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,𝑢₁,𝑣₁) with 𝑠=0, 𝑢²₁+𝑣²₁>0 and 𝑠>0, 𝑢²₁+𝑣²₁≥0, respectively. Clearly, system (4.1) is 𝐶¹ in (𝑠,𝑢₁,𝑣₁) as 0≤𝑠≪1, 𝑢²₁+𝑣²₁≪1, and 𝐶³ in (𝜖,𝜇), and has a solution 𝑠=𝑢₁=𝑣₁=0 as 𝜖=0, thus we can rescale 𝑠=𝜖²𝑠, 𝑢₁=𝜖²𝑢, 𝑣₁=𝜖²𝑣, such that system (4.1) is reformulated as 𝑀₁(𝜖)𝑀𝑇,𝜇+𝑂=0,₃(𝜖)𝑀𝑇,𝜇+𝑂=0,₄𝑇,𝜇+𝑂(𝜖)=0, (4.2) where 𝑂(𝜖) depends on 𝜖,𝑠,𝑢,𝑣 and 𝜇.

Now the following results are verified directly by the implicit function theorem.

Theorem 4.1. Assume (𝐴1)–(𝐴3) are satisfied, 𝑀₁(𝑇,𝜇₀)=𝑀₃(𝑇,𝜇₀)=𝑀₄(𝑇,𝜇₀)=0 and Rank(𝜕(𝑀₁,𝑀₃,𝑀₄)/𝜕𝜇)(𝑇,𝜇₀)=3 for 𝑙≥3 and 𝜇₀∈ℝ^𝑙, then, for any given 𝑟₀>0, 𝑠₀>0 and 𝜖₀>0 small enough, there exist a single-parameter family of codimension 3 surface Σ_𝜖 and a four-parameter family of codimension 3 surface Σ_{𝜖𝑢𝑣𝑠} near 𝜇₀ such that Σ₀=Σ_{0𝑢𝑣𝑠}={𝜇|𝜇=𝜇₀}, lim_𝑠→0Σ_{𝜖𝑢𝑣𝑠}=Σ_{𝜖𝑢𝑣0} for 0<𝜖<𝜖₀, 0≤𝑢²+𝑣²<𝑟²₀, 0<𝑠<𝑠₀, and system (2.1) has a symmetric 1-homoclinic orbit to 𝑂 as 𝜇∈Σ_𝜖, a 1-periodic orbit with approximate period 2𝑇−𝜆⁻¹log𝜖²𝑠 as 𝜇∈Σ_{𝜖𝑢𝑣𝑠}, and a 1-homoclinic orbit to a periodic orbit on the center manifold with 𝑢²₁(𝑡)+𝑣²₁(𝑡)=𝜖⁴(𝑢²₀+𝑣²₀) as 𝜇∈Σ_{𝜖𝑢0𝑣00}, Σ_{𝜖𝑢0𝑣00}−{𝜇₀}≠∅, 0<𝑢²₀+𝑣²₀<𝑟²₀. Moreover, these codimension 3 surfaces are either coincident, or tangent to each other at 𝜇=𝜇₀, and they have a common 3-dimensional normal space spanned by 𝑀_1𝜇(𝑇,𝜇₀),𝑀_3𝜇(𝑇,𝜇₀), and 𝑀_4𝜇(𝑇,𝜇₀) as 𝑙>3.

In order to study the existence of the 1-periodic orbit bifurcation surface with lower codimension, we reconsider (4.1). Let 𝑠=𝑒^{−(𝜆/𝜔)(2𝑘𝜋−𝛼)}, 𝛼∈[−(𝜋/2),(3𝜋/2)) and 𝐻 be the vector defined by the left hand side of the last two equations of (4.1), then, by using 𝑊≜det𝜕𝐻𝜕𝑢₁,𝑣₁^𝑢₁^=𝑣₁^=𝜖=0_𝑘=∞=−2+𝜌₃+𝜌₄+𝜌₃𝜌₄−2+𝜌₃+𝜌₄sin𝛼_{𝑢1=𝑣1=𝜖=0𝑘=∞}, (4.3) it follows that, for 𝛿>0 so small that |𝜌₃+𝜌₄|<0.1, |𝜌₃+𝜌₄+𝜌₃𝜌₄|<0.1, we have 𝑊≠0 for 𝛼∈[−(𝜋/4),(5𝜋/4)]. Therefore, there is a unique pair (𝑢₁,𝑣₁) which solves the last two equations as 𝑘>−(𝜔/𝜆𝜋)log𝜖 and 𝜖>0 is small enough.

Substituting this solution (𝑢₁,𝑣₁) into the first equation of (4.1), and using 𝑠=𝑂(𝜖²) for 𝑘>−(𝜔/𝜆𝜋)log𝜖, it produces 𝑀₁𝑇,𝜇+𝑂(𝜖)=0, (4.4) where the term 𝑂(𝜖) depends on 𝜖,𝑘,𝛼 and 𝜇.

At this stage, we have actually demonstrated the following existence theorem.

Theorem 4.2. Suppose (𝐴1)–(𝐴3) are satisfied, 𝑀₁(𝑇,𝜇₀)=0, (𝜕𝑀₁/𝜕𝜇)(𝑇,𝜇₀)≠0 as 𝑙≥1 and 𝜇₀∈ℝ^𝑙, then for 𝜖₀>0, 𝛿>0 small enough, 0<𝜖<𝜖₀, 𝑘>−(𝜔/𝜆𝜋)log𝜖 and 𝛼∈[−(𝜋/4),(5𝜋/4)], there exists a 3-parameter (resp., 2-parameter) family of codimension 1 surface Σ_𝜖𝑘𝛼 (resp., Σ_𝜖𝛼=Σ_𝜖∞𝛼) near 𝜇₀ satisfying Σ_0∞𝛼={𝜇₀}, such that system (2.1) has one 1-periodic orbit 𝐿(𝜖,𝑘,𝛼) (resp., 1-homoclinic orbit 𝐿(𝜖,∞,𝛼)) near Γ as 𝜇∈Σ_𝜖𝑘𝛼(resp., 𝜇∈Σ_𝜖𝛼), and 𝐿(𝜖,𝑘,𝛼) has an approximate period 2𝑇+((2𝑘𝜋−𝛼)/𝜔). In addition, if 𝑀₁(𝑇,𝜇)≠0 for any 𝜇∈ℝ^𝑙, then system (2.1) has not any 1-periodic orbit near Γ.

Remark 4.3. When confined in a neighborhood of the local center-manifold, the (𝑢,𝑣)-coordinates of 1-periodic or 1-homoclinic orbit guaranteed by Theorem 4.1 as 𝜇∈Σ_{𝜖𝑢𝑣𝑠} or 𝜇∈Σ_{𝜖𝑢0𝑣00} satisfying 𝑢²₁+𝑣²₁=𝑂(𝜖⁴), whereas, these coordinates of the corresponding orbits given in Theorem 4.2 as 𝜇∈Σ_𝜖𝑘𝛼 or 𝜇∈Σ_𝜖𝛼 have the scale 𝑂(𝜖²) in case 𝑀²₄(𝑇,𝜇₀)+𝑀²₃(𝑇,𝜇₀)≠0 and 𝑘>−(𝜔/𝜆𝜋)log𝜖.

5. Existence of Symmetric 1-Periodic Orbit

In this section, we turn to seek the existence of symmetric 1-periodic orbit. Note that, a 1-periodic orbit 𝐿 near Γ is symmetric if and only if 𝑅(𝐿∩Σ^𝑠)=𝐿∩Σ^𝑢, equivalently 𝐿∩Σ^𝑠=(𝑥₁,𝛿,𝑢₁,𝑣₁) and 𝐿∩Σ^𝑢=(𝛿,𝑦₀,𝑢₀,𝑣₀) should satisfy 𝑥₁=𝑦₀,𝑢₁=𝑣₀,𝑣₁=𝑢₀. (5.1) Due to (3.15), it is equivalent to 𝜌₁−𝜌₂𝑢=0,₁−𝑢₁𝜔sin𝜆log𝑠+𝑣₁𝜔cos𝜆log𝑠1+𝜌₄−𝜌₆𝑣=0,₁−𝑢₁𝜔cos𝜆log𝑠−𝑣₁𝜔sin𝜆log𝑠1+𝜌₃−𝜌₅=0. (5.2) Using (5.2) and rescaling 𝑢₁=𝜖𝑢, 𝑣₁=𝜖𝑣, 𝑠=𝑒^{−(𝜆/𝜔)(4𝑘𝜋+2𝜃)}=𝑂(𝜖²) for 𝑘>−(𝜔/2𝜆𝜋)log𝜖, 𝜃∈[0,2𝜋), system (4.1) reads as 𝑀₁𝑇,𝜇+𝑂(𝜖)=0,2𝑣+𝑀₃(𝜖)𝑀𝑇,𝜇+𝑂=0,₄𝑇,𝜇+𝑂(𝜖)=0. (5.3)

Solving the second equation of (5.3), we have 1𝑣=−2𝑀₃𝑇,𝜇+𝑂(𝜖). (5.4)

On the other hand, 𝐿∩Σ^𝑠 and 𝐿∩Σ^𝑢 are symmetric if and only if their middle point (𝑥(𝑇+(1/2)𝜏),𝑦(𝑇+(1/2)𝜏),𝑢(𝑇+(1/2)𝜏),𝑣(𝑇+(1/2)𝜏)) lies on the plane Fix(𝑅), that is, (5.1) is equivalent to 𝑥1𝑇+2𝜏1=𝑦𝑇+2𝜏1,𝑢𝑇+2𝜏1=𝑣𝑇+2𝜏, (5.5) where 𝑥1𝑇+2𝜏=𝛿√𝑠1+𝜌₁₁,𝑦1𝑇+2𝜏=𝛿√𝑠1+𝜌₂₁,𝑢1𝑇+2𝜏=𝑢₁cos𝜔𝜏2+𝑣₁sin𝜔𝜏21+𝜌₃₁+𝜌₅₁,𝑣1𝑇+2𝜏=−𝑢₁sin𝜔𝜏2+𝑣₁cos𝜔𝜏21+𝜌₄₁+𝜌₆₁, (5.6) and 𝜌_𝑖1=𝜌_𝑖1(𝑥₁,𝑦₁,𝑢₁,𝑣₁,𝜏,𝜖,𝜇) for 𝑖=1,…,6, 𝜌_𝑖1=𝑂(𝑥₁/√𝑠)+𝑂_𝑖(𝑦₁)+𝑂(𝑢²₁+𝑣²₁)=𝑂(𝛿), 𝑂_𝑖(𝑦₁)=𝑂(𝑦₁) for 𝑖=1,…,4, 𝜌_𝑗1=𝑂(𝑥₁𝑦₁)=𝑂(𝛿²𝑠) for 𝑗=5,6.

Explicitly, (5.5) can be formulated as 𝜌₁₁−𝜌₂₁𝑢=0,₁cos𝜔𝜏2+sin𝜔𝜏2−𝑣₁cos𝜔𝜏2−sin𝜔𝜏2+𝑂(𝛿)+𝑂𝛿²𝑠=0. (5.7) Due to (5.4) and (1/2)𝜔𝜏=2𝑘𝜋+𝜃, 𝜃∈[0,2𝜋), we can uniquely solve the second equation in (5.7) by 𝑢₁𝜖()=−cos𝜃−sin𝜃2()𝑀cos𝜃+sin𝜃³𝑇,𝜇(1+𝑂(𝛿))+𝑂𝜖² (5.8) as 𝜃≠(3/4)𝜋, (7/4)𝜋, and 𝛿, 𝜖 small enough.

Up to now, there are three equations 𝜌₁₁−𝜌₂₁𝑀=0,₁(𝜖)𝑀𝑇,𝜇+𝑂=0,₄𝑇,𝜇+𝑂(𝜖)=0 (5.9) left which should be fulfilled, where 𝑣₁=𝜖𝑣 and 𝑢₁ are given by (5.4) and (5.8).

Applying the implicit function theorem to (5.9), where 𝑂(𝜖) depends on 𝜖,𝜃,𝑘 and 𝜇, we derive the following result.

Theorem 5.1. Suppose that (𝐴1)–(𝐴3) hold, 𝑀₁(𝑇,𝜇₀)=0, 𝑀₄(𝑇,𝜇₀)=0 and 𝜕𝜌Rank¹¹−𝜌₂₁𝜕𝜇^{𝜖=0,𝜇=𝜇}_0𝑘=∞,𝜕𝑀₁,𝑀₄𝜕𝜇𝑇,𝜇₀=3 (5.10) as 𝜇₀∈ℝ^𝑙(𝑙≥3), there exist 𝜖₀>0, 𝜃₀>0 small enough, and a 3-parameter family of codimension 3 surface Σ_𝜖𝜃𝑘 near 𝜇₀, such that, system (2.1) has a symmetric 1-periodic orbit 𝐿 near Γ with approximate period 2𝑇+(1/𝜔)(4𝑘𝜋+2𝜃) as 𝜇∈Σ_𝜖𝜃𝑘 for 𝜖,𝜃,𝑘 satisfying 0<𝜖<𝜖₀, 𝜃∈[0,(3/4)𝜋−𝜃₀)∪((3/4)𝜋+𝜃₀, (7/4)𝜋−𝜃₀)∪((7/4)𝜋+𝜃₀,2𝜋), 𝑘>−(𝜔/2𝜆𝜋)log𝜖. Moreover, if there is no 𝜇₀∈ℝ^𝑙 satisfying 𝑀²₁𝑇,𝜇₀+𝑀²₄𝑇,𝜇₀+𝜌₁₁−𝜌₂₁²_{𝜖=0,𝜇=𝜇0𝑘=∞}=0, (5.11) then system (2.1) has no symmetric 1-periodic orbit in the small neighborhood of Γ.

Remark 5.2. Similarly, if we take 𝑠≜𝑒^−𝜆𝜏=𝑢₁=𝑣₁=0 in (3.15), then 𝑥₁=𝑦₀=𝑢₀=𝑣₀=𝑢₁=𝑣₁=0 (i.e., condition (5.1)), it means that there is a symmetric 1-homoclinic orbit to 𝑂 if and only if 𝑠=𝑢₁=𝑣₁=0 in Section 4.

Remark 5.3. In Theorem 4.2 (resp., Theorem 5.1), the geometric meaning of 𝑘 is that, confined in 𝑈₀, the 𝑢-𝑣 component of the above 1-periodic orbit bifurcated from Γ makes circle 𝑘 (resp., 2𝑘) times around the saddle-center.

Remark 5.4. From 𝑥₁/√𝑠=𝛿√𝑠(1+𝑂(𝛿)), 𝑠=𝑂(𝜖²) and the constitution of 𝜌₁₁ and 𝜌₂₁, it is easy to know that the necessary condition for 𝜌₁₁=𝜌₂₁ is 𝑂₁(𝑦₁)=𝑂₂(𝑦₁).

Acknowledgments

This paper is supported by NSF of China (11101370, 11126265, 11171085), the Research Fund for the Doctoral Program of Higher Education (20111420120006), and Zhejiang Provincial NSF of China (Y6100081).