The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues

Abstract

The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

1. Introduction and Setting of the Problem

In recent years, there is a large literature concerning the bifurcation problems of homoclinic and heteroclinic loops in dynamical systems (see [1–23] and the references therein). However, to the best of the authors’ knowledge, less attention has been devoted to the bifurcation of double homoclinic loops. Han and Bi [24] investigated the existence of homoclinic bifurcation curves and small and large limit cycles bifurcated from a double homoclinic loop under multiple parameter perturbations for general planar systems. Han and Chen [25] gave the number of limit cycles near double homoclinic loops under perturbations in planar Hamiltonian systems. Homburg and Knobloch [26] considered the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times in conservative and reversible systems. Morales et al. [27, 28] presented contracting and expanding Lorenz attractors through resonant double homoclinic loops. Lu [29] obtained codimension 2 bifurcations of twisted double homoclinic loops in higher dimensional systems. Ragazzo [30] investigated the stability of sets that were generalizations of the simple pendulum double homoclinic loop. In our recent work [31, 32], codimension 2 bifurcations of double homoclinic loops and codimension 3 bifurcations of nontwisted double homoclinic loops with resonant eigenvalues were studied. Concerning this topic, a more extensive list of references can be found in the references mentioned earlier. Generally speaking, when studying the problem of single homoclinic or heteroclinic bifurcation connecting hyperbolic equilibrium, the bifurcation is more complicated as Γ is twisted. Moreover, it is known that double homoclinic loops have higher codimension than a single homoclinic loop under the same conditions. Therefore, it will be more challenging and difficult to analyze the twisting bifurcations of double homoclinic loops.

In this paper, on the one hand, using the method which was originally established in [22, 23] and then improved in [9, 17, 21], and so forth, we study the bifurcations of double homoclinic loops with resonant eigenvalues under twisted cases. Besides, the method is more applicable and bifurcation equations obtained in this paper are easier to compute. On the other hand, frequently, the too many equivalent terms in the bifurcation equation will make the bifurcation equation more complex so that it is very difficult to analyze the bifurcation equation. Applying the method used by Homburg and Knobloch [26] to analyze the center-stable and center-unstable tangent bundles, we cannot only get a smooth coordinate transformation in the neighborhood of Γ small enough, but also make the bifurcation equation definite under an additional condition. Such strategy in dealing with the problems of bifurcations from homoclinic and heteroclinic loops is rarely used in the existing literatures. Therefore, a main feature in this paper is a combination of geometrical and analytical methods.

Motivated by these points, we will consider the following 𝐶^𝑟 system and its unperturbed system: ̇(𝑧),̇(𝑧),𝑧=𝑓+𝑔𝑧,𝜇𝑧=𝑓 (1)(2) where 𝑟 is large enough, 𝑧∈𝑅⁴,  𝜇∈𝑅^𝑙,  𝑙≥3,   0<|𝜇|≪1,  𝑓(0)=0,  𝑔(0,𝜇)=𝑔(𝑧,0)=0.

We make the following assumptions, which are shown in Figure 1. (𝐻₁)The linearization 𝐷𝑓(0) has simple real eigenvalues at the equilibrium 0∶−𝜌₂,−𝜌₁,𝜆₁,𝜆₂ satisfying −𝜌₂<−𝜌₁<0<𝜆₁<𝜆₂,𝜌₁=𝜆₁. (3)(𝐻₂) System (2) has double homoclinic loops Γ=Γ₁∪Γ₂,  Γ_𝑖={𝑧=𝑟_𝑖(𝑡)∶𝑡∈𝑅,𝑟_𝑖(±∞)=0} and dim(𝑇_{𝑟𝑖(𝑡)}𝑊^𝑠∩𝑇_{𝑟𝑖(𝑡)}𝑊^𝑢)=1, 𝑖=1,2, where 𝑊^𝑠 and 𝑊^𝑢 are the stable and unstable manifolds of 0, respectively.(𝐻₃) Let 𝑒^±_𝑖=lim_{𝑡→∓∞}(𝑟̇_𝑖(𝑡)/|𝑟̇_𝑖(𝑡)|), and 𝑒⁺_𝑖∈𝑇₀𝑊^𝑢, 𝑒⁻_𝑖∈𝑇₀𝑊^𝑠 unit eigenvectors corresponding to 𝜆₁ and −𝜌₁, respectively, and satisfying 𝑒⁺₁=−𝑒⁺₂, 𝑒⁻₁=−𝑒⁻₂.(𝐻₄)Span{𝑇_{𝑟𝑖(𝑡)}𝑊^𝑢,𝑇_{𝑟𝑖(𝑡)}𝑊^𝑠,𝑒⁺_𝑖}=𝑅⁴ as 𝑡≫1, and Span{𝑇_{𝑟𝑖(𝑡)}𝑊^𝑢,𝑇_{𝑟𝑖(𝑡)}𝑊^𝑠,𝑒⁻_𝑖}=𝑅⁴ as 𝑡≪−1, 𝑖=1,2.

Figure 1 

Double homoclinic loops Γ=Γ1∪Γ2.

As shown in Figure 1, under the hypotheses (𝐻₁)--(𝐻₄), we can see that the double homoclinic loops Γ are of codimension 3.

The single homoclinic loop in high-dimensional systems has been investigated by many authors (see [1, 2, 4–8, 13–15, 17, 21, 22] and the references therein). In this paper, we only focus on bifurcations of the large loop; that is, the double loops Γ=Γ₁∪Γ₂.

A nondegenerate homoclinic orbit is called a nontwisted homoclinic orbit if the unstable manifold 𝑊^𝑢 has an even number of half-twists along the homoclinic orbit, and is called a twisted homoclinic orbit if 𝑊^𝑢 has an odd number of half-twists along the homoclinic orbit; see more details in Deng [4]. A characterization for a twisted homoclinic orbit is that it arises from and tends to the equilibrium point from different sides of the unstable manifold. However, a nontwisted homoclinic orbit does so from the same side of the manifold. For the double homoclinic loop, Γ=Γ₁∪Γ₂, it is called double twisted if both Γ₁ and Γ₂ are twisted, single twisted if and only if one of them is twisted, and nontwisted otherwise.

The rest of this paper is organized as follows. In Section 2, the normal form in a neighborhood of the equilibrium small enough is established and the bifurcation equations are given. In Section 3, by bifurcation analysis, the results of twisting bifurcations and complete bifurcation diagrams are obtained under different cases. We end the paper with a conclusion in Section 4.

2. Normal Form and Bifurcation Equations

From the hypothesis (𝐻₁), we can see that we confine ourselves to consider the resonance taking place between the two principal eigenvalues 𝜆₁ and −𝜌₁. Moreover, the corresponding eigenvectors are also the tangent directions of the homoclinic orbits. For simplicity, we may choose a new parameter 𝜇=(𝛼,𝜈) such that 𝜌₁(𝜇)=𝜆₁(𝜈)+𝛼𝜆₁(𝜈), −1≪𝛼≪1.

Let 𝑈 be a neighborhood of 0 small enough. By analyzing the center-stable and center-unstable tangent bundles, Homburg and Knobloch [26] obtained that there is a smooth coordinate transformation, such that in 𝑈 system (1) takes the following form: 𝑥̇=𝑥𝜆₁(𝜈)(1)(𝑢)𝑂𝑦(𝑣),̇−()𝜆+𝑜+𝑂+𝑂𝑦=𝑦1+𝛼₁(𝜈)(1)(𝑣𝑂(𝑥)(𝑢,̇𝜆+𝑜+𝑂)(+𝑂))𝑢=𝑢₂(𝜈)+𝑜(1)+𝑥²𝐻₁,̇𝑥,𝑦,𝑣𝑣=𝑣−𝜌₂(𝜈)+𝑜(1)+𝑦²𝐻₂𝑥,𝑦,𝑢. (4) For the definiteness of the bifurcation equation, we make an additional assumption as follows:(𝐻₅)𝐻₁(𝑥,0,0)=0, 𝐻₂(0,𝑦,0)=0; that is, 𝐻₁𝑥,𝑦,𝑣=𝑎₁𝑥^𝑘1𝑦^𝑘2+𝑎₂𝑥^𝑘3𝑣^𝑘4+𝑎₃𝑥^𝑘5𝑦^𝑘6𝑣^𝑘7𝐻+h.o.t.,₂𝑥,𝑦,𝑢=𝑏₁𝑦^𝑙1𝑥^𝑙2+𝑏₂𝑦^𝑙3𝑢^𝑙4+𝑏₃𝑥^𝑙5𝑦^𝑙6𝑢^𝑙7+h.o.t., (5)where 𝑘_𝑖>𝜆₂/𝜆₁, 𝑖=1,3,5, 𝑘₂𝜌₁/𝜆₁>2, 𝑘₄𝜌₂/𝜆₁>2, (𝑘₆𝜌₁+𝑘₇𝜌₂)/𝜆₁>2. 𝑙_𝑖>𝜌₂/𝜌₁, 𝑖=1,3,5, 𝑙₂𝜆₁/𝜌₁>2, 𝑙₄𝜆₂/𝜌₁>2, (𝑙₆𝜆₁+𝑙₇𝜆₂)/𝜌₁>2.

By the same process as in [32], which is based on the analysis of the Poincaré return map defined on some local transversal section of the double homoclinic loop Γ, we obtain the bifurcation equations as follows: 𝑠₂=𝑤₁¹²⁻¹𝑠₁^1+𝛼−𝛿⁻¹𝑀¹₁𝑠𝜈+h.o.t.,₁=𝑤₂¹²⁻¹𝑠₂^1+𝛼+𝛿⁻¹𝑀¹₂𝜈+h.o.t. (6)

3. Bifurcation Analysis

Denote that 𝑤_𝑖¹²=Δ_𝑖|𝑤_𝑖¹²|. We say that Γ is nontwisted as Δ₁=Δ₂=1 and twisted as Δ₁=Δ₂=−1 or Δ₁Δ₂=−1. In this paper, we focus on the twisted bifurcations. It is easy to see that |𝑤_𝑖¹²| is the approximate expanding rate of the solution 𝑧¹_𝑖(𝑡) from 𝑇⁰_𝑖 to −𝑇¹_𝑖.

Case 1 (Δ₁=Δ₂=−1 (i.e., double twisted)). For convenience and simplicity, we use the following notations throughout Case 1: 𝑅¹₁=𝜈∶𝑀¹₁𝜈>0,𝑀¹₂|𝜈|,𝑅𝜈<0,≪1₁₂=𝜈∶𝑀¹₁𝜈<0,𝑀¹₂𝜈>0,𝑀¹₁𝜈=𝑂𝑀¹₂𝜈,|𝜈|,𝐷≪1₁=𝜈∈𝑅¹₂∶𝛿⁻¹𝑀¹₁𝜈^1+𝛼−𝑤₂¹²2^1/(1+𝛼)−1×𝛿⁻¹𝑀¹₂,𝐷𝜈>0for𝛼>0₂=𝜈∈𝑅¹₂∶−𝛿⁻¹𝑀¹₂𝜈^1+𝛼+𝑤₁¹²2^1/(1+𝛼)−1×𝛿⁻¹𝑀¹₁𝜈>0for𝛼>0. (7)

If (6) has solution 𝑠₁=𝑠₂=0, then we have 𝑀¹_𝑖𝜈+h.o.t.=0,𝑖=1,2. (8) If 𝑀¹_𝑖≠0, then there exists a codimension 1 surface Σ_𝑖 with a normal vector 𝑀¹_𝑖 at 𝜈=0, such that the 𝑖th equation of (6) has solution 𝑠₁=𝑠₂=0 as 𝜈∈Σ_𝑖 and |𝜈|≪1; that is, Γ_𝑖 is persistent. If rank(𝑀¹₁,𝑀¹₂)=2, then Σ₁₂=Σ₁∩Σ₂ is a codimension 2 surface with a normal plane span{𝑀¹₁,𝑀¹₂} such that (6) has solution 𝑠₁=𝑠₂=0 as 𝜈∈Σ₁₂ and |𝜈|≪1; equivalently, the large loop Γ=Γ₁∪Γ₂ is persistent.

Suppose that (6) has solution 𝑠₁=0,𝑠₂>0. Then, we have 𝑠₂=−𝛿⁻¹𝑀¹₁𝑤𝜈+h.o.t.,0=₂¹²⁻¹𝑠₂^1+𝛼+𝛿⁻¹𝑀¹₂𝜈+h.o.t. (9) Hence, we obtain the large 1-homoclinic orbit bifurcation surface equation 𝐻¹₂∶𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈+−𝛿⁻¹𝑀¹₁𝜈^1+𝛼+h.o.t.=0, (10) which is well defined at least in the region 𝑅¹₂ and has a normal vector 𝑀¹₂ as 𝛼>0 (resp., 𝑀¹₁ as 𝛼<0) at 𝜈=0 such that system (1) has a large 1-homoclinic loop Γ¹₂ near Γ as 𝜈∈𝐻¹₂. It also means that 𝐻¹₂ is tangent to Σ₂ as 𝛼>0 (resp., Σ₁ as 𝛼<0). When 𝜈∈𝐻¹₂, 𝛼>0, 𝑠₁=0, and 𝑠₂=−𝛿⁻¹𝑀¹₁𝜈+h.o.t., (6) indicates that 𝑠_2𝜈=−𝛿⁻¹𝑀¹₁𝑠+h.o.t.,_1𝜈=(1+𝛼)𝑤₂¹²⁻¹−𝛿⁻¹𝑀¹₁𝜈^𝛼𝑠_2𝜈+𝛿⁻¹𝑀¹₂+h.o.t. (11)(12) Using (11) ×(1+𝛼)(𝑤₂¹²)⁻¹(−𝛿⁻¹𝑀¹₁𝜈)^𝛼+ (12), we have 𝑠_1𝜈∣_𝐻¹₂=𝛿⁻¹𝑀¹₂+𝑂𝑀¹₁𝜈^𝛼. (13) So, 𝑠₁=𝑠₁(𝜈) increases along the direction of the gradient 𝑀¹₂ for 𝜈 near 𝐻¹₂.

Similarly, by setting 𝑠_𝑘=𝑒^{−𝜌1𝜏𝑘}, we can derive that 𝑠₁(𝜈) increases along the direction −𝑀¹₁ for 𝜈 near 𝐻¹₂ as 𝛼<0.

If (6) has solution 𝑠₁>0 and 𝑠₂=0, then we have 0=𝑤₁¹²⁻¹𝑠₁^1+𝛼−𝛿⁻¹𝑀¹₁𝑠𝜈+h.o.t.,₁=𝛿⁻¹𝑀¹₂𝜈+h.o.t. (14) Therefore, we obtain the existence of the other large 1-homoclinic orbit bifurcation surface equation 𝐻¹₁∶𝛿⁻¹𝑤₁¹²𝑀¹₁𝜈−𝛿⁻¹𝑀¹₂𝜈^1+𝛼+h.o.t.=0, (15) which is well defined at least in the region 𝑅¹₂ and has a normal vector 𝑀¹₁ as 𝛼>0 (resp., 𝑀¹₂ as 𝛼<0) at 𝜈=0 such that system (1) has a large 1-homoclinic loop Γ¹₁ near Γ as 𝜈∈𝐻¹₁. Thus, 𝐻¹₁ is also tangent to Σ₁ as 𝛼>0 (resp., Σ₂ as 𝛼<0). For 𝜈∈𝐻¹₁ and 𝛼>0, 𝑠₁=𝛿⁻¹𝑀¹₂𝜈+h.o.t., 𝑠₂=0, it follows from (6) that 𝑠_2𝜈=(1+𝛼)𝑤₁¹²⁻¹𝛿⁻¹𝑀¹₂𝜈^𝛼𝑠_1𝜈−𝛿⁻¹𝑀¹₁𝑠+h.o.t.,_1𝜈=𝛿⁻¹𝑀¹₂+h.o.t. (16)(17) Computing (17) ×(1+𝛼)(𝑤₁¹²)⁻¹(𝛿⁻¹𝑀¹₂𝜈)^𝛼+ (16), it leads to 𝑠_2𝜈∣_𝐻¹₁=−𝛿⁻¹𝑀¹₁+𝑂𝑀¹₂𝜈^𝛼. (18) So, 𝑠₂=𝑠₂(𝜈) increases along the direction −𝑀¹₁ for 𝜈 close to 𝐻¹₁.

By a similar procedure, we can show that 𝑠₂(𝜈) increases along the direction 𝑀¹₂ as 𝜈 is in the neighborhood of 𝐻¹₁ and 𝛼<0.

Summing up the previous analysis, we have the following theorem.

Theorem 1. Suppose that (𝐻₁)--(𝐻₅) hold; then, the following conclusions are true.(1)If 𝑀¹_𝑖≠0, then there exists a unique surface Σ_𝑖 with codimension 1 and normal vectors 𝑀¹_𝑖 at 𝜈=0, such that system (1) has a homoclinic loop near Γ_𝑖 if and only if 𝜈∈Σ_𝑖 and |𝜈|≪1. If rank(𝑀¹₁,𝑀¹₂)=2, then Σ₁₂=Σ₁∩Σ₂ is a codimension 2 surface and 0∈Σ₁₂ such that system (1) has a large loop consisting of two homoclinic orbits near Γ as 𝜈∈Σ₁₂ and |𝜈|≪1; that is, Γ is persistent.(2)In the region 𝑅¹₂, there exists a large 1-homoclinic orbit bifurcation surface 𝐻¹₂ which is tangent to Σ₂ (resp., Σ₁) at 𝜈=0 with the normal vector 𝑀¹₂ (resp., 𝑀¹₁) as 𝛼>0 (resp., 𝛼<0), and for 𝜈∈𝐻¹₂, system (1) has a unique large 1-homoclinic orbit near Γ. Furthermore, the unique large 1-homoclinic orbit near Γ becomes a large 1-periodic orbit when 𝜈 moves along the direction 𝑀¹₂ (resp., −𝑀¹₁) and nearby 𝐻¹₂ as 𝛼>0 (resp., 𝛼<0). Meanwhile, there exists another large 1-homoclinic orbit bifurcation surface 𝐻¹₁ which is tangent to Σ₁ (resp., Σ₂) at 𝜈=0 with the normal 𝑀¹₁ (resp., 𝑀¹₂) as 𝛼>0 (resp., 𝛼<0), and for 𝜈∈𝐻¹₁, system (1) has another one unique large 1-homoclinic orbit near Γ. Furthermore, the unique large 1-homoclinic orbit near Γ changes into a large 1-periodic orbit when 𝜈 shifts along the direction −𝑀¹₁ (resp., 𝑀¹₂) and near 𝐻¹₁ as 𝛼>0 (resp., 𝛼<0).

Lemma 2. Suppose that hypotheses (𝐻₁)--(𝐻₅) and 0<𝛼≪1 are valid; then, in addition to the large 1-homoclinic loop Γ¹₂, system (1) has exactly one simple large 1-periodic orbit near Γ for 𝜈∈𝐻¹₂∩𝐷₂. Moreover, the large 1-periodic orbit is persistent as 𝜈 changes in the neighborhood of 𝐻¹₂.

Proof. If 𝜈∈𝐻¹₂ and |𝜈|≪1, then we know that system (1) has one large 1-homoclinic loop Γ¹₂. Now, following (6), we have 𝑤₁¹²⁻¹𝑠₁^1+𝛼−𝛿⁻¹𝑀¹₁𝜈+h.o.t.^1+𝛼=𝑤₂¹²𝑠₁−𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈+h.o.t. (19) Let 𝑁₁(𝑠₁) and 𝐿₁(𝑠₁) be the left- and right-hand sides of (19), respectively. Then, by (10), we get 𝑁₁(0)=𝐿₁(0) as 𝜈∈𝐻¹₂. Moreover, 𝑁^′₁𝑠₁=(1+𝛼)²𝑤₁¹²⁻¹𝑤₁¹²⁻¹𝑠₁^1+𝛼−𝛿⁻¹𝑀¹₁𝜈+h.o.t.^𝛼𝑠^𝛼₁,𝐿^′₁𝑠₁=𝑤₂¹²+h.o.t. (20) Thus, 0=𝑁^′₁(𝑠₁)∣_𝑠1=0>𝐿^′₁(𝑠₁)∣_𝑠1=0. Therefore, there is an 𝑠₁̃, 0<𝑠₁̃≪1, such that for 0<𝑠₁<𝑠₁̃, 𝑁₁(𝑠₁)>𝐿₁(𝑠₁).
Let 𝑠₁=−𝛿⁻¹𝑀¹₂𝜈. Then, 𝑁₁𝑠₁=𝑤₁¹²⁻¹−𝛿⁻¹𝑀¹₂𝜈^1+𝛼−𝛿⁻¹𝑀¹₁𝜈+h.o.t.^1+𝛼,𝐿₁𝑠₁=𝑤₂¹²−𝛿⁻¹𝑀¹₂𝜈−𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈+h.o.t.=2−𝛿⁻¹𝑀¹₁𝜈^1+𝛼+h.o.t., (21) which means that 𝑁₁(𝑠₁)<𝐿₁(𝑠₁) as 𝜈∈𝐻¹₂∩𝐷₂. Then, it follows from 𝑁₁^′′(𝑠₁)<0 that 𝑁₁(𝑠₁)=𝐿₁(𝑠₁) has a unique solution 𝑠^∗₁ satisfying 0<𝑠₁̃<𝑠^∗₁≪1, which is shown in Figure 2. That is, system (1) has a unique large 1-periodic loop near Γ for 𝜈∈𝐻¹₂∩𝐷₂ and 0<|𝜈|≪1. The proof is complete.

Figure 2 

𝑁1(𝑠1) and 𝐿1(𝑠1) for 0≤𝑠1≪1, 𝜈∈𝐻12∩𝐷2.

Lemma 3. Suppose that hypotheses (𝐻₁)--(𝐻₅) and 0<𝛼≪1 hold. Then, system (1) has a saddle-node bifurcation surface of large 1-periodic orbit in the region 𝑅¹₂∩{𝜈∶|𝑀¹₁𝜈|=𝑜(|𝑀¹₂𝜈|)} as follows: 𝑆𝑁²∶−𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈=𝑤₁¹²⁻¹𝑤₁¹²𝑀¹₁𝜈(1+𝛼)²𝑀¹₂𝜈^{(1+𝛼)/𝛼}−𝛿⁻¹𝑀¹₁𝜈^1+𝛼−𝑤₂¹²𝑤₁¹²𝑀¹₁𝜈(1+𝛼)²𝑀¹₂𝜈^1/𝛼+h.o.t. (22)

Proof. It is easy to see that ℎ=𝑁₁(𝑠₁) is tangent to ℎ=𝐿₁(𝑠₁) at some point 𝑠₁, as shown in Figure 3; if 𝑁₁𝑠₁=𝐿₁𝑠₁,𝑁^′₁𝑠₁=𝐿^′₁𝑠₁, (23) then 𝑤₁¹²⁻¹𝑠₁^1+𝛼−𝛿⁻¹𝑀¹₁𝜈+h.o.t.^1+𝛼=𝑤₂¹²𝑠₁−𝛿⁻¹𝑤₂¹²𝑀¹₂()𝜈+h.o.t.,1+𝛼²𝑤₁¹²⁻¹𝑠^𝛼₁𝑤₁¹²⁻¹𝑠₁^1+𝛼−𝛿⁻¹𝑀¹₁𝜈+h.o.t.^𝛼=𝑤₂¹²+h.o.t. (24)(25) It is not difficult to see that (24) and (25) have a unique small positive solution 𝑠₁=𝑤₁¹²𝑀¹₁𝜈(1+𝛼)²𝑀¹₂𝜈^1/𝛼+h.o.t. (26) when 𝜈∈𝑅¹₂ and |𝑀¹₁𝜈|=𝑜(|𝑀¹₂𝜈|^1+𝛼). Substituting (26) into (24), we get the surface SN². It is easy to verify that the solution 𝑠₂ of (6) is positive as 𝑠₁ is given by (26) and 𝜈∈SN². Thus, the proof of the lemma is complete.

Figure 3 

𝑁1(𝑠1) and 𝐿1(𝑠1) for 0≤𝑠1≪1, 𝜈∈𝑆𝑁2.

Remark 4. Due to 𝑁₁^′′(𝑠₁), we see that the line 𝐿₁(𝑠₁) lies above the curve 𝑁₁(𝑠₁) as 𝜈∈SN². Moreover, 𝐿₁(0)=−𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈+h.o.t. decreases as rank(𝑀¹₁,𝑀¹₂)=2 and 𝜈 moves along the direction −𝑀¹₂; in this case, system (1) exhibits two large 1-periodic orbits. And as 𝜈 leaves SN² along the opposite direction, there is no large 1-periodic orbit.

Corollary 5. The bifurcation surface {0<𝛼≪1}×𝑆𝑁² and the region ({0<𝛼≪1}×𝑅¹₂)∩{(𝛼;𝜈)∶|𝑀¹₁𝜈|=𝑂(|𝑀¹₂𝜈|^1+𝛼),𝛼ln|𝑀¹₂𝜈|≪−1} have no intersection point.

Proof. Let 𝑤₁¹²|𝑀¹₁𝜈|=−𝑎^𝛼|𝑀¹₂𝜈|^1+𝛼, where 0<𝑎<𝛿⁻¹𝑒² is a given constant; then, in view of |𝑀¹₂𝜈|^𝛼≪1 and (1+𝛼)^1/𝛼→𝑒 as 𝛼→0, we know the right side of the expression SN² as follows: 𝑤₁¹²⁻¹𝑤₁¹²𝑀¹₁𝜈(1+𝛼)²𝑀¹₂𝜈^{(1+𝛼)/𝛼}−𝛿⁻¹𝑀¹₁𝜈^1+𝛼−𝑤₂¹²𝑤₁¹²𝑀¹₁𝜈(1+𝛼)²𝑀¹₂𝜈^1/𝛼<1+h.o.t.4𝑤₁¹²𝑒²𝑎^1+𝛼𝑀¹₂𝜈^1+𝛼−𝛿⁻¹𝑤₁¹²𝑎^𝛼𝑀¹₂𝜈^1+𝛼1+𝛼−𝑤₂¹²𝑤₁¹²𝑀¹₁𝜈(1+𝛼)²𝑀¹₂𝜈^1/𝛼=+h.o.t.4𝑤₁¹²𝑒^2−(1+𝛼)𝑀¹₂𝜈^(1+𝛼)2𝑎^1+𝛼−4𝛿⁻¹𝑒²𝑎^𝛼1+𝛼−𝑤₂¹²𝑤₁¹²𝑀¹₁𝜈(1+𝛼)²𝑀¹₂𝜈^1/𝛼+h.o.t.=𝑎^𝛼(1+𝛼)4𝑤₁¹²𝑒^2−(1+𝛼)𝑀¹₂𝜈^(1+𝛼)2𝑎−4𝛿⁻¹𝑒^21+𝛼−𝑤₂¹²𝑎𝑒²𝑀¹₂𝜈+h.o.t.<𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈, (27) and 𝛿⁻¹|𝑤₂¹²|𝑀¹₂𝜈 is the left side of the expression SN². Hence, the proof is complete.

Corollary 6. As 𝑤₁¹²𝑤₂¹²>1, the bifurcation surface {0<𝛼≪1}×𝑆𝑁² and the region ({0<𝛼≪1}×𝑅¹₂)∩{(𝛼;𝜈)∶|𝑀¹₁𝜈|=𝑂(|𝑀¹₂𝜈|^1+𝛼),−1≪𝛼ln|𝑀¹₂𝜈|<0} have no intersection point.

Proof. Still let 𝑤₁¹²|𝑀¹₁𝜈|=−𝑎^𝛼|𝑀¹₂𝜈|^1+𝛼, where 0<𝑎<𝛿⁻¹𝑒² is a given constant; then, due to |𝑀¹₂𝜈|^𝛼=1+𝑜(|𝑀¹₂𝜈|)<1, we know that 𝐿≜−𝛿⁻¹𝑤₁¹²𝑤₂¹²𝑀¹₂𝜈+𝑤₁¹²𝑤₂¹²𝑤₁¹²𝑀¹₁𝜈(1+𝛼)²𝑀¹₂𝜈^1/𝛼=𝑤₁¹²𝑤₂¹²𝑀¹₂𝜈−𝛿⁻¹+𝑎(1+𝛼)^−2/𝛼,𝑅≜𝑤₁^12−𝛼𝑤₁¹²𝑀¹₁𝜈(1+𝛼)²𝑀¹₂𝜈^{(1+𝛼)/𝛼}−𝛿⁻¹𝑤₁¹²𝑀¹₁𝜈^1+𝛼=)+h.o.t.(1+𝛼^{−2(1+𝛼)/𝛼}𝑎^1+𝛼𝑀¹₂𝜈^1+𝛼−𝛿⁻¹𝑎^𝛼𝑀¹₂𝜈^1+𝛼1+𝛼+h.o.t.>𝑎^𝛼(1+𝛼)𝑀¹₂𝜈𝛿⁻¹+𝑎(1+𝛼)^{−2/𝛼1+𝛼}+h.o.t. (28) As 𝑤₁¹²𝑤₂¹²>1, 𝑎>0 is a constant, and 𝛼>0 is small enough, it is easy to see that 𝑅>𝐿. This contradicts the expression of SN². The proof is complete.

By a similar analysis as in Lemmas 2 and 3, we can obtain the following results.

Lemma 7. Suppose that hypotheses (𝐻₁)--(𝐻₅) and 0<𝛼≪1 are valid; then, in addition to the large 1-homoclinic loop Γ¹₁, system (1) has exactly one simple large 1-periodic orbit near Γ for 𝜈∈𝐻¹₁∩𝐷₁. Moreover, the large 1-periodic orbit is persistent as 𝜈 changes in the neighborhood of 𝐻¹₁.

Lemma 8. Suppose that hypotheses (𝐻₁)--(𝐻₅) and 0<𝛼≪1 hold. Then, system (1) has a saddle-node bifurcation surface of large 1-periodic orbit in the region 𝑅¹₁∩{𝜈∶|𝑀¹₁𝜈|=𝑜(|𝑀¹₂𝜈|} as follows: 𝑆𝑁¹∶𝛿⁻¹𝑤₁¹²𝑀¹₁𝛿𝜈=⁻¹𝑀¹₂𝜈(1+𝛼)²(1+𝛼)²−1^1+𝛼−𝑤₁¹²𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈(1+𝛼)²−1^1/(1+𝛼)+h.o.t. (29)

Remark 9. Let 𝑁₂(𝑠₂)=[(𝑤₂¹²)⁻¹𝑠₂^1+𝛼+𝛿⁻¹𝑀¹₂𝜈+h.o.t.]^1+𝛼 and 𝐿₂(𝑠₂)=𝑤₁¹²𝑠₂+𝛿⁻¹𝑤₁¹²𝑀¹₁𝜈+h.o.t. Due to 𝑁₂^′′(𝑠₂)>0 and 𝐿₂(0)=𝛿⁻¹𝑤₁¹²𝑀¹₁𝜈+h.o.t., we claim that the line 𝐿₂(𝑠₂) lies above the curve 𝑁₂(𝑠₂) as 𝜈∈SN¹, and when rank(𝑀¹₁,𝑀¹₂)=2 and 𝜈 leaves SN¹ along the direction 𝑀¹₁, 𝐿₂(0) decreases; hence, system (1) has two large 1-periodic orbits, and when 𝜈 moves along the direction −𝑀¹₁, there is no large 1-periodic orbit.

Corollary 10. The bifurcation surface {0<𝛼≪1}×𝑆𝑁¹ and the region ({0<𝛼≪1}×𝑅¹₂)∩{(𝛼;𝜈)∶|𝑀¹₁𝜈|=𝑂(|𝑀¹₂𝜈|^1+𝛼),𝛼ln|𝑀¹₂𝜈|≪−1} have no intersection point.

Corollary 11. As 𝑤₁¹²𝑤₂¹²>1, the bifurcation surface {0<𝛼≪1}×𝑆𝑁¹ and the region ({0<𝛼≪1}×𝑅¹₂)∩{(𝛼;𝜈)∶|𝑀¹₁𝜈|=𝑂(|𝑀¹₂𝜈|^1+𝛼),−1≪𝛼ln|𝑀¹₂𝜈|<0} have no intersection point.

In the following we define open regions in the neighborhood of the origin of the 𝜈-space, which are shown in Figures 4 and 5.

Figure 4 

Bifurcation diagram in case Δ1=Δ2=−1, 0<𝛼≪1, −1≪𝛼ln|𝑀12𝜈|<0, 𝑤112𝑤212<1.

Figure 5 

Bifurcation diagram in case Δ1=Δ2=−1, 0<𝛼≪1, −1≪𝛼ln|𝑀12𝜈|<0, 𝑤112𝑤212>1.

(𝑅¹₂)₀ is bounded by SN¹ and SN², (𝑅¹₂)₁ is bounded by Σ₂ and 𝐻¹₂, (𝑅¹₂)₂ is bounded by 𝐻¹₂ and SN², (𝑅¹₂)^′₁ is bounded by SN¹ and SN², (𝑅¹₂)^′₂ is bounded by SN¹ and 𝐻¹₁, 𝑅¹₁ is bounded by Σ₁ and Σ₂, and (𝑅¹₂)₂^′′ is bounded by 𝐻¹₂ and 𝐻¹₁.

Now, the previous analysis is summarized in the following three theorems, as shown in Figures 4 and 5.

Theorem 12. Suppose that hypotheses (𝐻₁)--(𝐻₅) hold, 0<𝛼≪1, −1≪𝛼ln|𝑀¹₂𝜈|<0, 𝜈∈𝑅¹₂, and 𝑤₁¹²𝑤₂¹²<1. Then, system (1)(1)has exactly one simple large 1-periodic orbit near Γ as 𝜈∈(𝑅¹₂)₁;(2)has a unique double large 1-periodic orbit near Γ as 𝜈∈𝑆𝑁²;(3)has exactly two simple large 1-periodic orbits near Γ as 𝜈∈(𝑅¹₂)₂;(4)has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop near Γ as 𝜈∈𝐻¹₂∩𝐷₂;(5)does not have any large 1-periodic and large 1-homoclinic loop near Γ as 𝜈∈(𝑅¹₂)₀;(6)has exactly one simple large 1-periodic orbit near Γ as 𝜈∈(𝑅¹₂)^′₁;(7)has a unique double large 1-periodic orbit near Γ as 𝜈∈𝑆𝑁¹;(8)has exactly two simple large 1-periodic orbits near Γ as 𝜈∈(𝑅¹₂)^′₂;(9)has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop near Γ as 𝜈∈𝐻¹₁∩𝐷₁.

Theorem 13. Suppose that hypotheses (𝐻₁)--(𝐻₅) hold, 0<𝛼≪1, −1≪𝛼ln|𝑀¹₂𝜈|<0, 𝜈∈𝑅¹₂, and 𝑤₁¹²𝑤₂¹²>1. Then system (1)(1)has exactly one simple large 1-periodic orbit near Γ as 𝜈∈(𝑅¹₂)₁;(2)has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop near Γ as 𝜈∈𝐻¹₂∩𝐷₂;(3)has exactly two simple large 1-periodic orbits near Γ as 𝜈∈(𝑅¹₂)₂^′′;(4)has exactly one simple large 1-periodic orbit and one large 1-homoclinic loop near Γ as 𝜈∈𝐻¹₁∩𝐷₁;(5)has exactly one simple large 1-periodic orbit near Γ as 𝜈∈(𝑅¹₂)^′₁.

Remark 14. Suppose that hypotheses (𝐻₁)--(𝐻₅) hold, 0<𝛼≪1, 𝛼ln|𝑀¹₂𝜈|≪−1, 𝜈∈𝑅¹₂; then the conclusions of Theorem 13 are still true, and the bifurcation diagram is the same as Figure 5.

Denote by 𝐷²₁ the open region with boundaries Σ₁ and Σ₂, such that 𝐷²₁∩{𝜈∶𝑀¹₁𝜈>0,𝑀¹₂𝜈>0,|𝜈|≪1}≠∅. 𝐷¹₂ is the open region with boundaries Σ₂ and Σ₁, such that 𝐷¹₂∩{𝜈∶𝑀¹₁𝜈<0,𝑀¹₂𝜈<0,|𝜈|≪1}≠∅.

From the bifurcation equations (6), we have the following theorem.

Theorem 15. Suppose that hypotheses (𝐻₁)--(𝐻₅) hold, and 0<𝛼≪1. Then, system (1)(1)does not have any large 1-periodic orbit near Γ as 𝜈∈𝐷²₁;(2)does not have any large 1-periodic orbit near Γ as 𝜈∈𝐷¹₂;(3)does not have any large 1-periodic orbit near Γ as 𝜈∈𝑅¹₁.

Remark 16. For −1≪𝛼<0, by the same analysis, we can obtain the analogous results as those in Theorems 12–15. In fact, in the case we can reverse the time 𝑡 and change (𝑥,𝑦,𝑢,𝑣) to (𝑦,𝑥,𝑣,𝑢), and then we still get 0<𝛼≪1.

At last, we consider the case 𝛼=0. By a similar process to that of Section 3, we obtain the bifurcation equations −𝑤₁¹²⁻¹𝑠₁+𝑠₂+𝛿⁻¹𝑀¹₁𝑤𝜈+h.o.t.=0,₂₁₂₋₁𝑠₂−𝑠₁+𝛿⁻¹𝑀¹₂𝜈+h.o.t.=0. (30)

For the same reason as before, we have the following theorem.

Theorem 17. Suppose that (𝐻₁)--(𝐻₅), 𝛼=0, and rank(𝑀¹₁,𝑀¹₂)=2 hold; then,(1)as 𝜈∈𝑅¹₂, system (1) not only has a large 1-homoclinic orbit bifurcation surface 𝑤₂¹²𝑀¹₂𝜈−𝑀¹₁𝜈+h.o.t.=0, (31) with a normal vector (𝑤₂¹²𝑀¹₂−𝑀¹₁) at 𝜈=0, but also has another large 1-homoclinic orbit bifurcation surface 𝑤₁¹²𝑀¹₁𝜈−𝑀¹₂𝜈+h.o.t.=0; (32) with a normal vector (𝑤₁¹²𝑀¹₁−𝑀¹₂) at 𝜈=0,(2)as either 𝑤₁¹²𝑤₂¹²<1, 𝑀¹₂𝜈>𝑤₁¹²𝑀¹₁𝜈, and 𝑤₂¹²𝑀¹₂𝜈>𝑀¹₁𝜈 or 𝑤₁¹²𝑤₂¹²>1, 𝑀¹₂𝜈<𝑤₁¹²𝑀¹₁𝜈, and 𝑤₂¹²𝑀¹₂𝜈<𝑀¹₁𝜈, system (1) has a unique large 1-periodic orbit near Γ.

Case 2 (Δ₁=−1, Δ₂=1 (i.e., single twisted)). Similar to Case 1, for convenience and simplicity, we use the following notations: 𝑅¹₁=𝜈∶𝑀¹₁𝜈<0,𝑀¹₂𝜈>0,𝑀¹₂𝜈=𝑂𝑀¹₁𝜈,|𝜈|≪1,𝑅¹₂=𝜈∶𝑀¹₁𝜈<0,𝑀¹₂𝜈<0,𝑀¹₁𝜈=𝑂𝑀¹₂𝜈,|𝜈|≪1,𝐷²₁=𝜈∶𝑀¹₁𝜈>0,𝑀¹₂𝜈>0,|𝜈|≪1,𝐷¹₂=𝜈∶𝑀¹₁𝜈>0,𝑀¹₂𝜈<0,|𝜈|≪1. (33)
If (6) has a solution 𝑠₁=𝑠₂=0, then we have 𝑀¹_𝑖𝜈+h.o.t.=0,𝑖=1,2. (34) If 𝑀¹_𝑖≠0, then there exists a codimension 1 surface Σ_𝑖 with a normal vector 𝑀¹_𝑖 at 𝜈=0, such that the 𝑖th equation of (6) has solution 𝑠₁=𝑠₂=0 as 𝜈∈Σ_𝑖 and |𝜈|≪1; that is, Γ_𝑖 is persistent. If rank(𝑀¹₁,𝑀¹₂)=2, then Σ₁₂=Σ₁∩Σ₂ is a codimension 2 surface with a normal plane span{𝑀¹₁,𝑀¹₂} such that (6) has solution 𝑠₁=𝑠₂=0 as 𝜈∈Σ₁₂ and |𝜈|≪1; equivalently, the large loop Γ=Γ₁∪Γ₂ is persistent.
Suppose that (6) has solution 𝑠₁=0, 𝑠₂>0. Then, we have 𝑠₂=−𝛿⁻¹𝑀¹₁𝑤𝜈+h.o.t.,0=₂¹²⁻¹𝑠₂^1+𝛼+𝛿⁻¹𝑀¹₂𝜈+h.o.t. (35) Hence, we get the large 1-homoclinic orbit bifurcation surface equation 𝐻¹₂∶𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈+−𝛿⁻¹𝑀¹₁𝜈^1+𝛼+h.o.t.=0, (36) which is well defined at least in the region 𝑅¹₂ and has a normal vector 𝑀¹₂ as 𝛼>0 (resp., 𝑀¹₁ as 𝛼<0) at 𝜈=0 such that the system (1) has a large 1-homoclinic loop Γ¹₂ near Γ as 𝜈∈𝐻¹₂. It also means that 𝐻¹₂ is tangent to Σ₂ as 𝛼>0 (resp., Σ₁ as 𝛼<0). When 𝜈∈𝐻¹₂, 𝛼>0, 𝑠₁=0, and 𝑠₂=−𝛿⁻¹𝑀¹₁𝜈+h.o.t., (6) implies that 𝑠_2𝜈=−𝛿⁻¹𝑀¹₁𝑠+h.o.t.,_1𝜈=(1+𝛼)𝑤₂¹²⁻¹−𝛿⁻¹𝑀¹₁𝜈^𝛼𝑠_2𝜈+𝛿⁻¹𝑀¹₂+h.o.t. (37)(38) Using (37) ×(1+𝛼)(𝑤₂¹²)⁻¹(−𝛿⁻¹𝑀¹₁𝜈)^𝛼+ (38), we get 𝑠_1𝜈∣_𝐻¹₂=𝛿⁻¹𝑀¹₂+𝑂𝑀¹₁𝜈^𝛼. (39) So, 𝑠₁=𝑠₁(𝜈) increases along the direction of the gradient 𝑀¹₂ for 𝜈 near 𝐻¹₂.
Similarly, by setting 𝑠_𝑘=𝑒^{−𝜌1𝜏𝑘}, we can derive that 𝑠₁(𝜈) increases along the direction −𝑀¹₁ for near 𝐻¹₂ as 𝛼>0.
If (6) has solution 𝑠₁>0 and 𝑠₂=0, then we get 0=𝑤₁¹²⁻¹𝑠₁^1+𝛼−𝛿⁻¹𝑀¹₁𝑠𝜈+h.o.t.,₁=𝛿⁻¹𝑀¹₂𝜈+h.o.t. (40) Therefore, we obtain the existence of the other large 1-homoclinic orbit bifurcation surface equation 𝐻¹₁∶𝛿⁻¹𝑤₁¹²𝑀¹₁𝜈+𝛿⁻¹𝑀¹₂𝜈^1+𝛼+h.o.t.=0, (41) which is well defined at least in the region 𝑅¹₁ and has a normal vector 𝑀¹₁ as 𝛼>0 (resp., 𝑀¹₂ as 𝛼<0) at 𝜈=0 such that the system (1) has a large 1-homoclinic loop Γ¹₁ near Γ as 𝜈∈𝐻¹₁. Thus, 𝐻¹₁ is also tangent to Σ₁ as 𝛼>0 (resp., Σ₂ as 𝛼<0). For 𝜈∈𝐻¹₁ and 𝛼>0, 𝑠₁=𝛿⁻¹𝑀¹₂𝜈+h.o.t., 𝑠₂=0, it follows from (6) that 𝑠_2𝜈=(1+𝛼)𝑤₁¹²⁻¹𝛿⁻¹𝑀¹₂𝜈^𝛼𝑠_1𝜈−𝛿⁻¹𝑀¹₁𝑠+h.o.t.,_1𝜈=𝛿⁻¹𝑀¹₂+h.o.t. (42)(43) Computing (43) ×(1+𝛼)(𝑤₁¹²)⁻¹(𝛿⁻¹𝑀¹₂𝜈)^𝛼+ (42), it leads to 𝑠_2𝜈∣_𝐻¹₁=−𝛿⁻¹𝑀¹₁+𝑂𝑀¹₂𝜈^𝛼. (44) So, 𝑠₂=𝑠₂(𝜈) increases along the direction −𝑀¹₁ for 𝜈 close to 𝐻¹₁.
By a similar procedure, we can show that 𝑠₂(𝜈) increases along the direction 𝑀¹₂ as 𝜈 is in the neighborhood of 𝐻¹₁ and 𝛼<0.

Summarizing the previous analysis, we have the following theorem.

Theorem 18. Suppose that (𝐻₁)--(𝐻₅) hold; then, the following conclusions are true.(1)If 𝑀¹_𝑖≠0, then there exists a unique surface Σ_𝑖 with codimension 1 and normal vectors 𝑀¹_𝑖 at 𝜈=0, such that system (1) has a homoclinic loop near Γ_𝑖 if and only if 𝜈∈Σ_𝑖 and |𝜈|≪1. If rank(𝑀¹₁,𝑀¹₂)=2, then Σ₁₂=Σ₁∩Σ₂ is a codimension 2 surface and 0∈Σ₁₂ such that system (1) has a large loop consisting of two homoclinic orbits near Γ as 𝜈∈Σ₁₂ and |𝜈|≪1; that is, Γ is persistent.(2)In the region 𝑅¹₂, there exists a unique large 1-homoclinic orbit bifurcation surface 𝐻¹₂ which is tangent to Σ₂ (resp., Σ₁) at 𝜈=0 with the normal vector 𝑀¹₂ (resp., 𝑀¹₁) as 𝛼>0 (resp., 𝛼<0), and for 𝜈∈𝐻¹₂, system (1) has a unique large 1-homoclinic orbit near Γ. Furthermore, the unique large 1-homoclinic orbit near Γ becomes a large 1-periodic orbit when 𝜈 moves along the direction 𝑀¹₂ (resp., −𝑀¹₁) and nearby 𝐻¹₂ as 𝛼>0 (resp., 𝛼<0). In the region 𝑅¹₁, there exists another unique large 1-homoclinic orbit bifurcation surface 𝐻¹₁ which is tangent to Σ₁ (resp., Σ₂) at 𝜈=0 with the normal 𝑀¹₁ (resp., 𝑀¹₂) as 𝛼>0 (resp., 𝛼<0), and for 𝜈∈𝐻¹₁, system (1) has another one unique large 1-homoclinic orbit near Γ. Furthermore, the unique large 1-homoclinic orbit near Γ changes into a large 1-periodic orbit when 𝜈 shifts along the direction −𝑀¹₁ (resp., 𝑀¹₂) and nearby 𝐻¹₁ as 𝛼>0 (resp., 𝛼<0).

Lemma 19. Suppose that hypotheses (𝐻₁)--(𝐻₅) and 0<𝛼≪1 are valid; system (1) has only a unique large 1-homoclinic orbit Γ¹₂ near Γ for 𝜈∈𝐻¹₂ and 0<|𝜈|≪1.

Proof. If 𝜈∈𝐻¹₂ and |𝜈|≪1, from Theorem 18, we know that system (1) has one large 1-homoclinic loop Γ¹₂. Now, following (6), we have 𝑤₁¹²⁻¹𝑠₁^1+𝛼−𝛿⁻¹𝑀¹₁𝜈+h.o.t.^1+𝛼=𝑤₂¹²𝑠₁−𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈+h.o.t. (45) Let 𝑁₁(𝑠₁) and 𝐿₁(𝑠₁) be the left- and right-hand sides of (45), respectively. Then, by (36), we get 𝑁₁(0)=𝐿₁(0) as 𝜈∈𝐻¹₂. Moreover, 𝑁^′₁𝑠₁=(1+𝛼)²𝑤₁¹²⁻¹𝑤₁¹²⁻¹𝑠₁^1+𝛼−𝛿⁻¹𝑀¹₁𝜈+h.o.t.^𝛼𝑠^𝛼₁,𝐿^′₁𝑠₁=𝑤₂¹²+h.o.t. (46) Thus, 𝑁^′₁(𝑠₁)≤0<𝐿^′₁(𝑠₁), and 𝑁₁^′′(𝑠₁)<0. Therefore, 𝑁₁(𝑠₁)<𝐿₁(𝑠₁) for 𝑠₁≠0, as shown in Figure 6.
Therefore, system (1) has only a unique large 1-homoclinic orbit near Γ for 𝜈∈𝐻¹₂ and 0<|𝜈|≪1.

Figure 6 

𝑁1(𝑠1) and 𝐿1(𝑠1) for 0≤𝑠1≪1, 𝜈∈𝐻12.

Remark 20. Due to 𝑁₁^′′(𝑠₁)<0, we can see that the line 𝑁₁(𝑠₁) lies under the curve 𝐿₁(𝑠₁) as 𝜈∈𝐻¹₂. Moreover, 𝐿₁(0)=−𝛿⁻¹𝑤₂¹²𝑀¹₂𝜈+h.o.t. decreases as rank(𝑀¹₁,𝑀¹₂)=2 and 𝜈 moves along the direction 𝑀¹₂; in this case, system (1) has a unique large 1-periodic orbit. And as 𝜈 leaves 𝐻¹₂ along the opposite direction, there is no large 1-periodic orbit.

By a similar analysis as in Lemma 19, we can obtain the following results.

Lemma 21. Suppose that hypotheses (𝐻₁)--(𝐻₅) and 0<𝛼≪1 are valid; system (1) has only a unique large 1-homoclinic orbit Γ¹₁ near Γ for 𝜈∈𝐻¹₁ and 0<|𝜈|≪1.

Remark 22. Let 𝑁₂(𝑠₂)=[(𝑤₂¹²)⁻¹𝑠₂^1+𝛼+𝛿⁻¹𝑀¹₂𝜈+h.o.t.]^1+𝛼 and 𝐿₂(𝑠₂)=𝑤₁¹²𝑠₂+𝛿⁻¹𝑤₁¹²𝑀¹₁𝜈+h.o.t. Due to 𝑁₂^′′(𝑠₂)>0 and 𝐿₂(0)=𝛿⁻¹𝑤₁¹²𝑀¹₁𝜈+h.o.t., we claim that the line 𝐿₂(𝑠₂) lies under the curve 𝑁₂(𝑠₂) as 𝜈∈𝐻¹₁, and when 𝑟𝑎𝑛𝑘(𝑀¹₁,𝑀¹₂)=2 and 𝜈 leaves 𝐻¹₁ along the direction −𝑀¹₁, 𝐿₂(0) increases; hence, system (1) has a unique large 1-periodic orbit, and when 𝜈 moves along the direction 𝑀¹₁, there is no large 1-periodic orbit.

In the following we define open regions in the neighborhood of the origin of the 𝜈-space, which are shown in Figure 7.

Figure 7 

Bifurcation diagram in case Δ1=−1, Δ2=1, 0<𝛼≪1.

(𝑅¹₂)₀ is bounded by Σ₁ and 𝐻¹₂, (𝑅¹₂)₁ is bounded by Σ₂ and 𝐻¹₂, (𝑅¹₁)₁ is bounded by 𝐻¹₁ and Σ₂, (𝑅¹₁)₀ is bounded by Σ₁ and 𝐻¹₁, 𝐷¹₂ is bounded by Σ₁ and Σ₂, and 𝐷²₁ is bounded by Σ₂ and Σ₁.

From the previous analysis, we obtain the following three theorems, as shown in Figure 7.

Theorem 23. Suppose that hypotheses (𝐻₁)--(𝐻₅) hold, 0<𝛼≪1, 𝜈∈𝑅¹₂. Then, system (1)(1)does not have any large 1-periodic orbit and large 1-homoclinic loop near Γ as 𝜈∈(𝑅¹₂)₀;(2)has a unique large 1-homoclinic loop near Γ as 𝜈∈𝐻¹₂;(3)has a unique large 1-periodic orbit near Γ as 𝜈∈(𝑅¹₂)₁.

Theorem 24. Suppose that hypotheses (𝐻₁)--(𝐻₅) hold, 0<𝛼≪1, 𝜈∈𝑅¹₁. Then, system (1)(1)has no large 1-periodic orbit and large 1-homoclinic loop near Γ as 𝜈∈(𝑅¹₁)₀;(2)has a unique large 1-homoclinic loop near Γ as 𝜈∈𝐻¹₁;(3)has a unique large 1-periodic orbit near Γ as 𝜈∈(𝑅¹₁)₁.

From the bifurcation equations, we have the following theorem easily.

Theorem 25. Suppose that hypotheses (𝐻₁)--(𝐻₅) hold, and 0<𝛼≪1. Then, system (1)(1)has no large 1-periodic orbit near Γ as 𝜈∈𝐷²₁;(2)has no large 1-periodic orbit near Γ as 𝜈∈𝐷¹₂.

Remark 26. For −1≪𝛼<0, by the same analysis, we can obtain the analogous results as those in Theorems 23–25. In fact, in the case we can reverse the time 𝑡 and change (𝑥,𝑦,𝑢,𝑣) to (𝑦,𝑥,𝑣,𝑢), and then we still get 0<𝛼≪1.

At last, we consider the case 𝛼=0. By a similar process to that of Section 3, we obtain the bifurcation equations −𝑤₁¹²⁻¹𝑠₁+𝑠₂+𝛿⁻¹𝑀¹₁𝑤𝜈+h.o.t.=0,₂₁₂₋₁𝑠₂−𝑠₁+𝛿⁻¹𝑀¹₂𝜈+h.o.t.=0. (47)

For the same reason as before, we have the following theorem.

Theorem 27. Suppose that (𝐻₁)--(𝐻₅), 𝛼=0, and rank(𝑀¹₁,𝑀¹₂)=2 hold; then,(1)as 𝜈∈𝑅¹₂, system (1) has a unique large 1-homoclinic orbit bifurcation surface 𝑤₂¹²𝑀¹₂𝜈−𝑀¹₁𝜈+h.o.t.=0, (48) with a normal vector (𝑤₂¹²𝑀¹₂−𝑀¹₁) at 𝜈=0;(2)as 𝜈∈𝑅¹₁, system (1) also has a unique large 1-homoclinic orbit bifurcation surface 𝑤₁¹²𝑀¹₁𝜈−𝑀¹₂𝜈+h.o.t.=0, (49) with a normal vector (𝑤₁¹²𝑀¹₁−𝑀¹₂) at 𝜈=0;(3)as 𝑀¹₂𝜈<𝑤₁¹²𝑀¹₁𝜈 and 𝑤₂¹²𝑀¹₂𝜈>𝑀¹₁𝜈, system (1) has a unique large 1-periodic orbit near Γ.

Remark 28. As Γ₁ is nontwisted and Γ₂ is twisted, we can obtain similar conclusions as shown in Case 2.

4. Conclusion

This paper is devoted to investigating the twisting bifurcations of double homoclinic loops with resonant eigenvalues in 4-dimensional systems. We give asymptotic expressions of the bifurcation surfaces and their relative positions, describe the existence regions of large 1-periodic orbits near Γ in Lemmas 2 and 7, and obtain the sufficient conditions for the existence or nonexistence of saddle-node bifurcation surfaces in Lemmas 3 and 8. More importantly, the complete bifurcation diagrams are given under different cases in Figures 4, 5, and 7. According to our analysis, when Γ is double twisted, in Theorems 12–15, we obtain one bifurcation diagram with saddle-node bifurcation surfaces of large 1-periodic orbit when 𝑤₁¹²𝑤₂¹²<1 and the other bifurcation diagram without saddle-node bifurcation surfaces when 𝑤₁¹²𝑤₂¹²>1. When Γ is single twisted, in Theorems 23–25, we obtain another bifurcation diagram. Compared with the nontwisted cases in Zhang et al. [32], our paper shows completely different results and bifurcation diagrams. It is worthy to be mentioned that the restriction on the dimension is not essential, the method used in this paper can be extended to higher dimensional systems without any difficulty, and the same conclusions can be deduced under the same hypotheses. Furthermore, we mention some problems for future study. (1)  𝑊^𝑠 or 𝑊^𝑢 is inclination flip on one of the double homoclinic loops. (2) Both 𝑊^𝑠 and 𝑊^𝑢 are inclination flips on the double homoclinic loops and so on. But the difficulty of these problems will increase with adding codimension of the double homoclinic loops.

Acknowledgments

The research was supported by National Natural Science Foundation of China (nos. 11001041, 11101170, 11202192, 11271065 and 10926105), SRFDP (no. 200802001008), and the State Scholarship Fund of the China Scholarship Council (nos. 2011662521, and 2011842509).