Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

Abstract

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.

1. Introduction and Hypotheses

The study of homoclinic flip bifurcations is comprehensively developed from the last two decades with the beginning work of Yanagida (1987) for homoclinic-doubling bifurcations. Generally there exist two kinds of homoclinic flips, namely the orbit flips and the inclination flips corresponding to nonprincipal homoclinic orbits or critically twisted homoclinic orbits, respectively. Kisaka et al. in [1, 2] and Naudot in [3] studied some cases of codimension two inclination flips; Morales and Pacifico in [4] and Naudot in [5] considered the orbit flips cases, while Homburg and Krauskopf in [6] proposed several unfoldings of the resonant homoclinic flip bifurcations around the central codimension-three point (the organizing centre) in parameter space to study the qualitative structure of bifurcation curves on a sphere and also that of Oldeman et al. in [7] by a numerical investigation with some software into these bifurcations in a specific three-dimensional vector field.

Recently, Zhang et al. in [8–10] studied a kind of multiple flips homoclinic resonant bifurcation and got the existence of some saddle-node bifurcations and homoclinic-doubling bifurcations. Meanwhile Geng et al. in [11], Lu et al. in [12], and Liu in [13] discussed, respectively, a heterodimensional cycle flip or accompanied by transcritical bifurcation; they found the double and triple periodic orbit bifurcations and gave also some coexistence conditions for homoclinic orbits and periodic orbits.

As mentioned in [6, 7], due to the break of three genericity conditions, there are many complicated homoclinic flips cases to study. In this paper, we confine our attention to a principal eigenvalue resonance of one orbit flip and one inclination flip homoclinic bifurcation. Compared with the above-mentioned work, our subject is very challenging and difficult because of the stronger degeneracy and the higher codimension. By constructing specifically a local active coordinate in a small tubular neighborhood of homoclinic orbit, we establish a regular map and then combine it with a singular map defined by the approximation solutions of system to build Poincaré return map (see also [14]). We obtain the existence of several 1-periodic orbit, 1-homoclinic orbit, and double 1-periodic orbits, as well as some bifurcation surfaces with the analysis of the bifurcation equation.

We first consider a 𝐶^𝑟 system 𝑧̇=𝑓(𝑧)+𝑔𝑧,𝜇 (1) and its unperturbed system 𝑧̇=𝑓(𝑧), (2) where 𝑟≥3, 𝑧∈ℝ⁴, 𝜇∈ℝ^𝑙, 𝑙≥4, 0<|𝜇|≪1, 𝑓(0)=0, and 𝑔(0,𝜇)=𝑔(𝑧,0)=0. Suppose that the linearization D𝑓(0) has four simple real eigenvalues 𝜆₁, 𝜆₂, −𝜌₁, and −𝜌₂ with 𝜆₂>𝜆₁>0>−𝜌₁>−𝜌₂. Accordingly, the stable manifold 𝑊^𝑠 and the unstable manifold 𝑊^𝑢 are both two-dimensional. Let 𝑊^𝑠𝑠 and 𝑊^𝑢𝑢 be the strong stable manifold and the strong unstable manifold of the saddle 𝑧=𝑂, respectively. Assume further that system (2) has a homoclinic orbit Γ={𝑧=𝑟(𝑡)∶𝑡∈ℝ,𝑟(±∞)=0}. Hereinafter, our arguments will spread based on the following three hypotheses. (H1) Resonance. 𝜆₁(𝜇)=𝜌₁(𝜇), |𝜇|≪1, where 𝜆₁(0)=𝜆₁ and 𝜌₁(0)=𝜌₁. (H2) Orbit Flip. Define 𝑒⁺=lim_{𝑡→−∞}𝑟(̇𝑡)/|𝑟(̇𝑡)|; 𝑒⁻_𝑠=lim_𝑡→+∞𝑟(̇𝑡)/|𝑟(̇𝑡)|; then 𝑒⁺∈𝑇₀𝑊^𝑢 and 𝑒⁻_𝑠∈𝑇₀𝑊^𝑠𝑠 are unit eigenvectors corresponding to 𝜆₁ and −𝜌₂, respectively, where 𝑇₀𝑊^𝑢 (resp., 𝑇₀𝑊^𝑠𝑠) is the tangent space of the corresponding manifold 𝑊^𝑢 (resp., 𝑊^𝑠𝑠) at the saddle 𝑧=𝑂. (H3) Inclination Flip. Let 𝑒⁺_𝑢 and 𝑒⁻ be the unit eigenvectors corresponding to 𝜆₂ and −𝜌₁, respectively, and lim_𝑡→+∞𝑇_𝑟(𝑡)𝑊^𝑢,𝑇_𝑟(𝑡)𝑊^𝑠,𝑒⁺_𝑢=ℝ⁴,lim_{𝑡→−∞}𝑇_𝑟(𝑡)𝑊^𝑢,𝑇_𝑟(𝑡)𝑊^𝑠,𝑒⁻=ℝ⁴. (3)

Remark 1. Hypothesis (H2) is called an orbit flip because homoclinic orbit trends from the weak unstable manifold toward the strong stable manifold. Hypothesis (H3) means an inclination flip for its equivalence to 𝑇_𝑟(𝑡)𝑊^𝑢⟶span𝑒⁻_𝑠,𝑒⁺,𝑇_𝑟(𝑡)𝑊^𝑠⟶span𝑒⁻_𝑠,𝑒⁺. (4)

2. Poincaré Return Map

This section treats mainly the establishment of Poincaré return map with two steps. To begin we first need to transform system (1) into a normal form in some neighborhood of the origin 𝑂.

It is well known that there are always two 𝐶^𝑟 and 𝐶^𝑟−1 transformations successively, also by the stable (or unstable) manifold theorem in [15], to straighten the local manifolds 𝑊^𝑠_loc and 𝑊^𝑢_loc as 𝑊^𝑠_loc={𝑧∈𝑈,𝑥=𝑢=0} and 𝑊^𝑢_loc={𝑧∈𝑈,𝑦=𝑣=0}, respectively, 𝑊^𝑠𝑠_loc={𝑧∈𝑈,𝑥=𝑦=𝑢=0} (resp., 𝑊^𝑢𝑢_loc={𝑧∈𝑈,𝑥=𝑦=𝑣=0}); see [8–10]. Notice that now Γ∩𝑊^𝑢_loc={𝑧∈𝑈,𝑢=𝑢(𝑥),𝑦=𝑣=0} and Γ∩𝑊^𝑠_loc=𝑊^𝑠𝑠_loc, where 𝑧=(𝑥,𝑦,𝑢,𝑣)∈ℝ⁴ and 𝑢(0)=𝑢^′(0)=0. Thus, system (1) can be changed to a 𝐶^𝑟−2 form in the neighborhood 𝑈 as follows: 𝑥̇=𝜆₁𝜇𝜇𝑥(𝑢𝑂𝑥+𝑎𝑥𝑣+𝑜(|𝑥𝑣|)+𝑂)²𝑣𝑦,̇+𝑂𝑦=−𝜌₁𝜇𝜇𝑦(𝑣𝑂(𝑢,̇𝜆+𝑏𝑥𝑣+𝑜(|𝑥𝑣|)+𝑂)𝑥𝑦𝑣+𝑂)𝑢=₂𝜇+𝑐𝜇𝑥𝑣+𝑜(|𝑥𝑣|)𝑢+𝑥²𝐻₁,̇𝑥,𝑦,𝑣𝑣=−𝜌₂𝜇+𝑑𝜇𝑥𝑣+𝑜(|𝑥𝑣|)𝑣+𝑦²𝐻₂𝑥,𝑦,𝑢, (5) where 𝐻₁(𝑥,0,0)=0, 𝐻₂(0,𝑦,0)=0, 𝜆₁(0)=𝜆₁, 𝜆₂(0)=𝜆₂, 𝜌₁(0)=𝜌₁, and 𝜌₂(0)=𝜌₂. 𝑎(𝜇), 𝑏(𝜇), 𝑐(𝜇), and 𝑑(𝜇) are parameters depending on 𝜇.

Owing to the above straightness of the invariant manifolds, it is easy to find some moment 𝑇, such that 𝑟(−𝑇)={𝛿,0,𝛿_𝑢,0} and 𝑟(𝑇)={0,0,0,𝛿} for some sufficiently small 𝛿 and |𝛿_𝑢|=𝑂(𝛿²) with {(𝑥,𝑦,𝑢,𝑣)∶|𝑥|,|𝑦|,|𝑢|,|𝑣|<2𝛿}⊂𝑈. Wherefore one can choose 𝑆₀=(𝑇)∶|𝑥|,𝑦,|𝑢|,|𝑣|𝑆𝑧=𝑧<2𝛿⊂𝑈,₁=𝑧=𝑧(−𝑇)∶|𝑥|,𝑦,|𝑢|,|𝑣|<2𝛿⊂𝑈, (6) as the cross-sections of Γ at 𝑡=𝑇 and 𝑡=−𝑇, respectively. Let 𝜏 be the time going from 𝑞₀(𝑥₀,𝑦₀,𝑢₀,𝑣₀)∈𝑆₀ to 𝑞₁(𝑥₁,𝑦₁,𝑢₁,𝑣₁)∈𝑆₁ and the Silnikov time 𝑠=𝑒^{−𝜆1(𝜇)𝜏}, with the help of the linear approximation solutions of system (5) (see [8–10]); we have thereby 𝑥₀=𝑥(𝑇)=𝑠𝑥₁+𝑂𝑥²₁𝑦₀𝑠²,𝑦ln𝑠₁=𝑦(𝑇+𝜏)=𝑠^{𝜌1(𝜇)/𝜆1(𝜇)}𝑦₀+𝑂𝑥₁𝑦²₀𝑠²,𝑢ln𝑠₀=𝑢(𝑇)=𝑠^{𝜆2(𝜇)/𝜆1(𝜇)}𝑢₁+𝑂𝑥₁𝑦₀𝑢₁𝑠^{(𝜆2(𝜇)/𝜆1(𝜇))+1},𝑣ln𝑠₁=𝑣(𝑇+𝜏)=𝑠^{𝜌2(𝜇)/𝜆1(𝜇)}𝑣₀+𝑂𝑥₁𝑦₀𝑣₀𝑠^{(𝜌2(𝜇)/𝜆1(𝜇))+1}ln𝑠, (7) which give explicitly the definition of the local transition map 𝐹₀∶𝑆₀→𝑆₁∶𝑞₀↦𝑞₁; see Figure 1(a).

(a) 𝐹0: 𝑆0→𝑆1

(b) 𝐹1: 𝑆1→𝑆0

Figure 1 

Transition maps.

In the following part we construct the map 𝐹₁∶𝑆₁→𝑆₀. Firstly consider the linear variational system 𝑧̇=𝐷𝑓(𝑟(𝑡))𝑧 (8) and its adjoint system 𝑧̇=−𝐷𝑓(𝑟(𝑡))^∗𝑧. (9)

Lemma 2. There exists a fundamental solution matrix 𝑍(𝑡)=(𝑧₁(𝑡), 𝑧₂(𝑡), 𝑧₃(𝑡), 𝑧₄(𝑡)) of system (8) satisfying 𝑍()=𝑤−𝑇¹¹𝑤₂₁0𝑤₄₁𝑤₁₂00𝑤₄₂𝑤₁₃𝑤₂₃1𝑤₄₃000𝑤₄₄,𝑍(𝑇)=00𝑤₃₁0𝑤₁₄0𝑤₃₂110𝑤₃₃001𝑤₃₄0, (10) where 𝑧₁(𝑡)∈(𝑇_𝑟(𝑡)𝑊^𝑢)^𝑐∩(𝑇_𝑟(𝑡)𝑊^𝑠)^𝑐, 𝑧₂(𝑡)=−𝑟(̇𝑡)/|𝑟̇^𝑦(𝑇)|∈𝑇_𝑟(𝑡)𝑊^𝑢∩𝑇_𝑟(𝑡)𝑊^𝑠, 𝑧₃(𝑡)∈𝑇_𝑟(𝑡)𝑊^𝑢, 𝑧₄(𝑡)∈𝑇_𝑟(𝑡)𝑊^𝑠 and 𝑤₁₂𝑤₂₁𝑤₃₁𝑤₄₄≠0, 𝑤₂₃<0. Moreover, |𝑤_1𝑖𝑤⁻¹₁₂|≪1, 𝑖≠2, |𝑤_3𝑖𝑤⁻¹₃₁|≪1,𝑖≠1, |𝑤_4𝑖𝑤⁻¹₄₄|≪1, 𝑖≠4 for 𝑇→+∞.

Proof. Notice that the tangent subspace 𝑇_{𝛾(−𝑇)}𝑊^𝑢 is invariant and 𝑊^𝑢𝑢_loc∩𝑈 is straightened to be 𝑢 axis; it is possible to choose 𝑧₃(−𝑇)=(0,0,1,0) since 𝑧₃(𝑡)∈𝑇_𝛾(𝑡)𝑊^𝑢. While for 𝑤₃₁≠0, it is because lim_𝑡→+∞𝑇_𝛾(𝑡)𝑊^𝑢=span{𝑒⁺,𝑒⁻_𝑠} and 𝑧₃(𝑇)∈𝑇_𝛾(𝑇)𝑊^𝑢 corresponding to 𝑥 axis.
As to 𝑧_𝑖(−𝑇) or 𝑧_𝑖(𝑇), 𝑖=1,2,4, one may refer to [8, 9] for the similar proof, but we omit the details here.

Remark 3. The matrix (𝑍⁻¹(𝑡))^∗ is a fundamental solution matrix of system (9). We denote it as Φ(𝑡)=(𝜙₁(𝑡),𝜙₂(𝑡),𝜙₃(𝑡),𝜙₄(𝑡))=(𝑍⁻¹(𝑡))^∗; then 𝜙₁(𝑡)∈(𝑇_𝑟(𝑡)𝑊^𝑢)^𝑐∩(𝑇_𝑟(𝑡)𝑊^𝑠)^𝑐 is bounded and tends to zero exponentially as |𝑡|→+∞ due to <𝜙_𝑖(𝑡),𝑧_𝑖(𝑡)≥1 and 𝑧_𝑖(𝑡) tends exponentially to infinity.

In fact (𝑧₁(𝑡),𝑧₂(𝑡),𝑧₃(𝑡),𝑧₄(𝑡)) can be regarded as a new local coordinate system along Γ. So we may make a coordinate change as 𝑠(𝑡)(𝑡)(𝑡)𝑁(𝑡)=𝑟+𝑍=𝑟+𝑧₁(𝑡)𝑛₁+𝑧₃(𝑡)𝑛₃+𝑧₄(𝑡)𝑛₄, (11) where 𝑁=𝑁(𝑡)=(𝑛₁(𝑡),0,𝑛₃(𝑡),𝑛₄(𝑡))^∗. Note that the new 𝑠(𝑡) should satisfy system (1); that is, 𝑠(̇𝑡)(𝑠(𝑡𝑠(𝑡)(𝑟(𝑡)(𝑡)𝑁)𝑟(𝑡)(𝑡).=𝑓))+𝑔,𝜇=𝑓+𝑍+𝑔+𝑍𝑁,𝜇 (12) An asymptotic expansion with respect to 𝑟(𝑡) shows that 𝑛̇_𝑖=𝜙^∗_𝑖(𝑡)𝑔_𝜇(𝑟(𝑡),0)𝜇+𝑂𝜇²+𝑂|𝑁|²+𝑂𝜇|𝑁|,𝑖=1,3,4. (13) Via integrating both sides from −𝑇 to 𝑇 of this equation, one can finally obtain 𝑛_𝑖(𝑇)=𝑛_𝑖(−𝑇)+𝑀_𝑖𝜇+h.o.t.,𝑖=1,3,4, (14) where 𝑀_𝑖=∫^𝑇_−𝑇𝜙^∗_𝑖(𝑡)𝑔_𝜇(𝑟(𝑡),0)d𝑡,𝑖=1,3,4 are the Melnikov vectors. And further 𝑀₁=∫^𝑇_−𝑇𝜙^∗₁(𝑡)𝑔_𝜇(𝑟(𝑡),0)d𝑡=∫^+∞_−∞𝜙^∗₁(𝑡)𝑔_𝜇(𝑟(𝑡),0)d𝑡.

Equation (14) defines exactly the map 𝐹₁∶𝑆₁→𝑆₀; 𝑁(−𝑇)↦𝑁(𝑇) under the new coordinate system; see Figure 1(b).

In order to combine 𝐹₀ and 𝐹₁ into the Poincaré return map, we still need to establish a relationship between the original and the new coordinate systems. In doing so, recall that 𝑧(𝑡)=𝑟(𝑡)+𝑍(𝑡)𝑁(𝑡); then by taking time 𝑡=𝑇 and 𝑡=−𝑇, respectively, together with 𝑧(𝑇)=𝑞_2𝑗(𝑥_2𝑗,𝑦_2𝑗,𝑢_2𝑗,𝑣_2𝑗)∈𝑆₀, 𝑧(−𝑇)=𝑞_2𝑗+1(𝑥_2𝑗+1,𝑦_2𝑗+1,𝑢_2𝑗+1,𝑣_2𝑗+1)∈𝑆₁ and 𝑁_2𝑗(𝑇)=(𝑛_2𝑗,1,0,𝑛_2𝑗,3,𝑛_2𝑗,4), 𝑁_2𝑗+1(−𝑇)=(𝑛_2𝑗+1,1,0,𝑛_2𝑗+1,3, 𝑛_2𝑗+1,4), 𝑗=0,1,2,…, we obtain immediately the following formulas: 𝑛_2𝑗,1=𝑢_2𝑗−𝑤₃₃𝑤⁻¹₃₁𝑥_2𝑗,𝑛_2𝑗,3=𝑤⁻¹₃₁𝑥_2𝑗,𝑛_2𝑗,4=𝑦_2𝑗−𝑤₁₄𝑢_2𝑗+𝑤₁₄𝑤₃₃−𝑤₃₂𝑤⁻¹₃₁𝑥_2𝑗,𝑛_2𝑗+1,1=𝑤⁻¹₁₂𝑦_2𝑗+1−𝑤₄₂𝑤⁻¹₁₂𝑤⁻¹₄₄𝑣_2𝑗+1,𝑛_2𝑗+1,3=𝑢_2𝑗+1−𝛿_𝑢−𝑤₁₃𝑤⁻¹₁₂𝑦_2𝑗+1+𝑤₁₃𝑤₄₂𝑤⁻¹₁₂−𝑤₄₃𝑤⁻¹₄₄𝑣_2𝑗+1,𝑛_2𝑗+1,4=𝑤⁻¹₄₄𝑣_2𝑗+1,𝑥_2𝑗+1≈𝛿,𝑣_2𝑗≈𝛿. (15)

With all of the above, the Poincaré return map is given as 𝐹=𝐹₁∘𝐹₀. Therefore, the associated successor function 𝐺(𝑠, 𝑢₁, 𝑦₀)=(𝐺₁, 𝐺₃, 𝐺₄)=𝐹(𝑞₀)−𝑞₀ is 𝐺₁=𝑤⁻¹₁₂𝑠^{𝜌1(𝜇)/𝜆1(𝜇)}𝑦₀−𝑢₁𝑠^{𝜆2(𝜇)/𝜆1(𝜇)}+𝑤₃₃𝑤⁻¹₃₁𝛿𝑠−𝑤₄₂𝑤⁻¹₁₂𝑤⁻¹₄₄𝛿𝑠^{𝜌2(𝜇)/𝜆1(𝜇)}+𝑀₁𝐺𝜇+h.o.t.,₃=𝑢₁−𝛿_𝑢−𝑤₁₃𝑤⁻¹₁₂𝑠𝑦₀−𝑤⁻¹₃₁+𝑤𝛿𝑠₁₃𝑤₄₂𝑤⁻¹₁₂−𝑤₄₃𝑤⁻¹₄₄𝛿𝑠^{𝜌2(𝜇)/𝜆1(𝜇)}+𝑀₃𝐺𝜇+h.o.t.,₄=𝑤⁻¹₄₄𝛿𝑠^{𝜌2(𝜇)/𝜆1(𝜇)}−𝑦₀+𝑤₁₄𝑠^{𝜆2(𝜇)/𝜆1(𝜇)}𝑢₁+𝑤₃₂−𝑤₁₄𝑤₃₃𝑤⁻¹₃₁𝛿𝑠+𝑀₄𝜇+h.o.t.. (16)

3. Bifurcation Results

From the definition of the Silnikov time 𝑠=𝑒^{−𝜆1(𝜇)𝜏}, we know that a solution with 𝑠>0 of (16) or equivalently 𝜏>0 corresponds to a periodic orbit near Γ and a solution with 𝑠=0 of (16) or equivalently 𝜏=+∞ corresponds to a homoclinic orbit near Γ. It is enough to study the solutions of the successor function 𝐺(𝑠,𝑢₁,𝑦₀)=0 for bifurcation analysis. Consider for concision that we omit the dependence on parameter 𝜇 from now on in the exponent notation.

From 𝐺₃=0 and 𝐺₄=0, there are 𝑢₁=𝛿_𝑢+𝑤⁻¹₃₁𝛿𝑠−𝑀₃𝜇+𝑂𝑠𝑦₀+𝑂𝑠^𝜌2/𝜆1,𝑦₀=𝑤₃₂−𝑤₃₃𝑤₁₄𝑤⁻¹₃₁𝛿𝑠+𝑀₄𝜇+𝑂𝑠^𝜌2/𝜆1+𝑂𝑠^𝜆2/𝜆1. (17)

Put 𝑢₁ and 𝑦₀ into 𝐺₁=0; we have 𝐹𝑠,𝜇≡𝑀₄𝜇𝑠−𝑤₄₂𝑤⁻¹₄₄𝛿𝑠^𝜌2/𝜆1+𝑤₁₂𝑤₃₃𝑤⁻¹₃₁𝛿𝑠−𝛿_𝑢𝑤₁₂𝑠^𝜆2/𝜆1+𝑤₁₂𝑀₃𝜇𝑠^𝜆2/𝜆1+𝑤₃₂−𝑤₃₃𝑤₁₄𝑤⁻¹₃₁𝛿𝑠²+𝑤⁻¹₄₄𝛿𝑠^{(𝜌1+𝜌2)/𝜆1}−𝑤₁₂𝑤⁻¹₃₁𝛿𝑠^{1+(𝜆2/𝜆1)}+𝑤₁₂𝑀₁𝜇+𝑤₁₄𝛿_𝑢𝑠^{(𝜌1+𝜆2)/𝜆1}+𝑤₁₃𝑀₄𝜇𝑠^{(𝜆1+𝜆2)/𝜆1}+h.o.t.=0, (18) which is the bifurcation equation. Furthermore, for 𝑤₃₃≠0, det𝐺_𝑄=0𝜇=0=𝑤₃₃𝑤⁻¹₃₁𝛿00−𝛿𝑤⁻¹₃₁𝑤10₃₂−𝑤₃₃𝑤₁₄𝑤⁻¹₃₁𝛿0−1≠0, (19) where 𝐺=𝜕(𝐺₁,𝐺₃,𝐺₄)/𝜕𝑄(𝑠,𝑢₁,𝑦₀). Implicit function theorem reveals that (16) has a unique solution as 𝑠=𝑠𝜇,𝑢₁=𝑢₁𝜇,𝑦₀=𝑦₀𝜇, (20) with 𝑠(0)=0, 𝑢₁(0)=0, and 𝑦₀(0)=0. It means that system (1) has a unique periodic orbit as 𝑠>0 or a unique homoclinic orbit as 𝑠=0, and they cannot coexist.

Theorem 4. Suppose that 𝑀₁≠0 and 𝑤₃₃≠0 are true; then system (1) has a unique 1-periodic orbit near Γ for 𝑤₃₁𝑤₃₃𝑀₁𝜇<0 or has a unique 1-homoclinic orbit Γ_𝜇 near Γ as 𝜇∈𝐻=^1Δ{𝜇∶𝑀₁𝜇+ℎ.𝑜.𝑡.=0}, and they do not coexist.

Proof. Clearly 𝐹(𝑠,𝜇)=0 has a small positive solution 𝑠=−𝛿⁻¹𝑤⁻¹₃₃𝑤₃₁𝑀₁𝜇+h.o.t., for 𝑤₃₁𝑤₃₃𝑀₁𝜇<0, and has a zero solution 𝑠=0 for 𝜇∈𝐻¹ which is a codimension-one hypersurface.

In the following part we restrict our attention on the case 𝑤₃₃=0 for 2𝜆₁>𝜆₂>𝜌₂. Define 𝑅^𝑘𝑙_𝑖𝑗=𝜇∶(−1)^𝑘𝑤₁₂^𝑙𝑀_𝑖𝜇𝑀_𝑗,𝐸𝜇>0_𝑗𝑘𝑙=𝜇∶(−1)^𝑘𝑤₁₂^𝑙𝑤₄₂𝑤₄₄𝑀_𝑗𝜇>0, (21) where 𝑘,𝑙=0,1,𝑖=1,3,𝑗=1,3,4.

In order to well solve (18), we rewrite it into two parts, namely, a line 𝑊=𝑃(𝑠,𝜇) and a curve 𝑊=𝑄(𝑠,𝜇) with respect to 𝑠: 𝑃𝑠,𝜇=𝑀₄𝜇𝑠+𝑤₁₂𝑀₁𝑄𝜇+h.o.t.,𝑠,𝜇=𝑤₄₂𝑤⁻¹₄₄𝛿𝑠^𝜌2/𝜆1+𝛿_𝑢𝑤₁₂𝑠^𝜆2/𝜆1−𝑤₁₂𝑀₃𝜇𝑠^𝜆2/𝜆1−𝑤₃₂𝑤⁻¹₃₁𝛿𝑠²−𝑤⁻¹₄₄𝛿𝑠^{(𝜌1+𝜌2)/𝜆1}+𝑤₁₂𝑤⁻¹₃₁𝛿𝑠^{(𝜆1+𝜆2)/𝜆1}+h.o.t.. (22)

Then there are firstly the following conclusions based on an analysis of the relative position of the line 𝑊=𝑃(𝑠,𝜇) and the curve 𝑊=𝑄(𝑠,𝜇).

Lemma 5. Suppose that 2𝜆₁>𝜆₂>𝜌₂, 𝑤₃₃=0, and 𝑤₄₂≠0 hold; then 𝐹(𝑠,𝜇)=0 has a unique small positive solution 𝑠̃∈(0,𝑠^∗) for 𝜇∈𝐸₁⁰¹, where 𝑠^∗=[𝑤₄₄𝑤⁻¹₄₂𝛿(2𝑤₁₂𝑀₁𝜇+𝑀₄𝜇)]^𝜆1/𝜌2.

Proof. It is clear that 𝑃0,𝜇=𝑤₁₂𝑀₁𝑃𝜇+h.o.t.,𝑄0,𝜇=0,_′𝑠,𝜇=𝑀₄𝑄𝜇+h.o.t.,_′𝑠,𝜇=𝜌₂𝜆₁⁻¹𝑤₄₂𝑤⁻¹₄₄𝛿𝑠^{(𝜌2/𝜆1)−1}+h.o.t.. (23) Therefore, the line 𝑊=𝑃(𝑠,𝜇) intersects the curve 𝑊=𝑄(𝑠,𝜇) at a unique point 𝑠̃ for 𝜇∈𝐸₁⁰¹. Notice that 𝑄(𝑠^∗,𝜇)=2𝑤₁₂𝑀₁𝜇+𝑀₄𝜇+h.o.t.>𝑤₁₂𝑀₁𝜇+𝑀₄𝜇𝑠^∗+h.o.t.=𝑃(𝑠^∗,𝜇), so 𝑠̃∈(0,𝑠^∗).

Theorem 6. Suppose that 2𝜆₁>𝜆₂>𝜌₂, 𝑤₃₃=0, and 𝑤₄₂≠0 hold; then system (1) has exactly a unique (resp., not any) 1-periodic orbit for 𝜇∈𝐸₁⁰¹ (resp., 𝜇∈𝐸₁¹¹∩𝑅⁰¹₁₄).

Proof. From Lemma 5, we know that 𝐹(𝑠,𝜇)=0 has exactly a unique small positive solution for 𝜇∈𝐸₁⁰¹ which corresponds exactly to a 1-periodic orbit of system (1). Moreover, 𝐹(𝑠,𝜇)=0 does not have any small positive solutions for 𝜇∈𝐸₁¹¹∩𝑅⁰¹₁₄.

Theorem 7. Suppose that 2𝜆₁>𝜆₂>𝜌₂, 𝑤₃₃=0, and 𝑤₄₂≠0 hold; then, for 𝜇∈𝐸₁¹¹∩𝑅¹¹₁₄ and Rank (𝑀₁,𝑀₄)=2, system (1) has a unique double 1-periodic orbit near Γ on the bifurcation surface 𝑆𝑁¹∶𝑤₁₂𝑀₁𝜌𝜇+²−𝜆₁𝜌₂𝜆₁𝑤₄₄𝜌₂𝑤₄₂𝛿^{𝜆1/(𝜌2−𝜆1)}×𝑀₄𝜇^{𝜌2/(𝜌2−𝜆1)}+ℎ.𝑜.𝑡.=0, (24) which has a normal vector 𝑀₁ at 𝜇=0. The corresponding double positive zero point is 𝑠_∗=𝜆₁𝑤₄₄𝑀₄𝜇𝜌₂𝑤₄₂𝛿^{𝜆1/(𝜌2−𝜆1)}+ℎ.𝑜.𝑡. (25) as 𝜇∈𝐸₄⁰⁰ (see Figure 2(a)).

(a) Double 1-periodic orbit

(b) Two 1-periodic orbits

(c) No 1-periodic orbit

Figure 2 

Rank (𝑀1,𝑀4)=2 and 𝜇∈𝐸111∩𝑅1114.

Proof. We know that the existence of a double 1-periodic orbit corresponds to the equations 𝑃(𝑠,𝜇)=𝑄(𝑠,𝜇), 𝑃^′(𝑠,𝜇)=𝑄^′(𝑠,𝜇), and 𝑃^′′(𝑠,𝜇)≠𝑄^′′(𝑠,𝜇), that is, 𝑀₄𝜇𝑠+𝑤₁₂𝑀₁𝜇=𝑤₄₂𝑤⁻¹₄₄𝛿𝑠^𝜌2/𝜆1+𝛿_𝑢𝑤₁₂𝑠^𝜆2/𝜆1−𝑤₁₂𝑀₃𝜇𝑠^𝜆2/𝜆1−𝑤₃₂𝑤⁻¹₃₁𝛿𝑠²−𝑤⁻¹₄₄𝛿𝑠^{(𝜌1+𝜌2)/𝜆1}+𝑤₁₂𝑤⁻¹₃₁𝛿𝑠^{(𝜆1+𝜆2)/𝜆1}𝑀+h.o.t.,₄𝜌𝜇=²𝜆₁𝑤₄₂𝑤⁻¹₄₄𝛿𝑠^{(𝜌2/𝜆1)−1}+𝜆₂𝜆₁𝛿_𝑢𝑤₁₂𝑠^{(𝜆2/𝜆1)−1}−𝜆₂𝜆₁𝑤₁₂𝑀₃𝜇𝑠^{(𝜆2/𝜆1)−1}−2𝑤₃₂𝑤⁻¹₃₁−𝜌𝛿𝑠₁+𝜌₂𝜆₁𝑤⁻¹₄₄𝛿𝑠^𝜌2/𝜆1+𝜆₁+𝜆₂𝜆₁𝑤₁₂𝑤⁻¹₃₁𝛿𝑠^𝜆2/𝜆1+h.o.t.,0≠𝜌₂𝜌₂−𝜆₁𝑤₄₂𝑤⁻¹₄₄𝛿𝑠^{(𝜌2/𝜆1)−2}+𝜆₂𝜆₂−𝜆₁𝛿_𝑢𝑤₁₂𝑠^{(𝜆2/𝜆1)−2}−𝜆₂𝜆₂−𝜆₁𝑤₁₂𝑀₃𝜇𝑠^{(𝜆2/𝜆1)−2}−2𝜆²₁𝑤₃₂𝑤⁻¹₃₁𝛿+𝜌₂𝜌₁+𝜌₂𝑤⁻¹₄₄𝛿𝑠^{(𝜌2/𝜆1)−1}+𝜆₂𝜆₁+𝜆₂𝑤₁₂𝑤⁻¹₃₁𝛿𝑠^{(𝜆2/𝜆1)−1}+h.o.t., (26) having solutions. Indeed, the second equation of (26) permits the double positive zero point 𝑠_∗ as 𝜇∈𝐸₄⁰⁰. Putting it into the first equation of (26), there is 𝑀₄𝜇𝜆₁𝑤₄₄𝑀₄𝜇𝜌₂𝑤₄₂𝛿^{𝜆1/(𝜌2−𝜆1)}+𝑤₁₂𝑀₁𝜇=𝑤₄₂𝑤⁻¹₄₄𝛿𝜆₁𝑤₄₄𝑀₄𝜇𝜌₂𝑤₄₂𝛿^{𝜌2/(𝜌2−𝜆1)}+h.o.t.. (27) Then 𝑆𝑁¹ exists for 𝜇∈𝐸₁¹¹∩𝑅¹¹₁₄.

From the above proof, we see that, when 𝑀₄𝜇>0 and 𝑤₁₂𝑀₁𝜇<0, the line 𝑊=𝑃(𝑠,𝜇) has a positive slope lying under the curve 𝑊=𝑄(𝑠,𝜇) when 𝑤₄₂𝑤₄₄>0, so if 𝑤₁₂𝑀₁𝜇 increases, the line must intersect the curve at two sufficiently small positive points, which can be equal to the existence of two 1-periodic orbits of system (1). For 𝑀₄𝜇<0, 𝑤₄₂𝑤₄₄<0, and 𝑤₁₂𝑀₁𝜇>0, the arguments are similar. So we have immediately a complement of Theorem 7.

Corollary 8. Assume that the hypotheses of Theorem 7 are valid, system (1) then has two (resp., not any) 1-periodic orbits near Γ when 𝜇 lies on the side of 𝑆𝑁¹ which points to the direction (sgn𝑤₁₂𝑤₄₂𝑤₄₄)𝑀₁ (resp., in the opposite direction) (see Figures 2(b) and 2(c)).

As Melnikov functions generally play an important role in bifurcation study, the following theorem shows also the existence of some double 1-periodic orbits relying on the investigation of 𝑀_𝑖=0 for 𝑖=1,3,4.

Theorem 9. Suppose 2𝜆₁>𝜆₂>𝜌₂, 𝑤₃₃=0, and 𝑤₄₂≠0 are valid; then the following applies.(1)For 𝑀₁=0 or 𝑀²₁+𝑀²₃=0, system (1) has exactly one 1-homoclinic orbit and one (resp., not any) 1-periodic orbit near Γ and they (resp., do not) coexist as 𝜇∈𝐸₄⁰⁰ (resp., 𝜇∈𝐸₄¹⁰).(2)For 𝑀₃=0, system (1) has exactly one (resp., not any) 1-periodic orbit near Γ as 𝜇∈𝐸₁⁰¹ (resp., 𝜇∈𝐸₁¹¹∩𝑅⁰¹₁₄). system (1) has a unique double 1-periodic orbit near Γ as 𝜇∈𝐸₁¹¹∩𝑅¹¹₁₄ and 𝑅𝑎𝑛𝑘(𝑀₁,𝑀₄)=2. Accordingly, the double 1-periodic orbit bifurcation surface is 𝑆𝑁¹∶𝑤₁₂𝑀₁𝜇+((𝜌₂−𝜆₁)𝑤₄₂𝛿/𝜆₁𝑤₄₄)(𝜆₁𝑤₄₄𝑀₄𝜇/𝜌₂𝛿𝑤₄₂)^{𝜌2/(𝜌2−𝜆1)}+ℎ.𝑜.𝑡.=0 with a normal vector 𝑀₁ at 𝜇=0, and it may generate two 1-periodic orbits when 𝜇 lies in the direction (sgn𝑤₄₂𝑤₄₄𝑤₁₂)𝑀₁ of 𝑆𝑁¹ and no such a 1-periodic orbit in the opposite direction.(3)For 𝑀₄=0 or 𝑀²₃+𝑀²₄=0, system (1) has only one (resp. not any) 1-periodic orbit near Γ as 𝜇∈𝐸₁⁰¹ (resp., 𝜇∈𝐸₁¹¹).(4)For 𝑀²₁+𝑀²₄=0, system (1) does not have any 1-periodic orbit near Γ.(5)For 𝑀²₁+𝑀²₃+𝑀²₄=0, system (1) has only one 1-homoclinic orbit near Γ.

Proof. When 𝑀₁=0 or 𝑀₁=𝑀₃=0, 𝐹(𝑠,𝜇)=𝑠(𝑀₄𝜇−𝑤₄₂𝑤⁻¹₄₄𝛿𝑠^{(𝜌2−𝜆1)/𝜆1}−𝛿_𝑢𝑤₁₂𝑠^{(𝜆2−𝜆1)/𝜆1}+𝑤₁₂𝑀₃𝜇𝑠^{(𝜆2−𝜆1)/𝜆1}+𝑤₃₂𝑤⁻¹₃₁𝛿𝑠+𝑤⁻¹₄₄𝛿𝑠^𝜌2/𝜆10+h.o.t.)= has two solutions 𝑠₁=0 and 𝑠₂=(𝑤₄₄𝑀₄𝜇/𝑤₄₂𝛿)^{𝜆1/(𝜌2−𝜆1)}+h.o.t. for 𝜇∈𝐸₄⁰⁰ which correspond to a 1-periodic orbit, and a 1-homoclinic orbit respectively. Thus, (1) is true.
The result of the cases (2) is exactly the same as that of Theorem 7.
If 𝑀₄=0 or 𝑀₃=𝑀₄=0, 𝐹(𝑠,𝜇)=0 has only a positive solution 𝑠₃=(𝑤₄₄𝑤₁₂𝑀₁𝜇/𝛿𝑤₄₂)^𝜆1/𝜌2+h.o.t. for 𝜇∈𝐸₁⁰¹, which means system (1) has a 1-periodic orbit. Then (3) is valid.
In case of 𝑀²₁+𝑀²₄=0, it is clear that 𝐹(𝑠,𝜇)=0 does not have any small nonnegative solutions, so system (1) does not have, any 1-homoclinic orbits or 1-periodic orbits.
The last conclusion is obvious. Thereby, the proof is complete.

Remark 10. Notice that, in Theorem 9  (1) and (5), 𝐹(𝑠,𝜇)=0 has a solution 𝑠=0, which means that system (1) has a codimension-1 1-homoclinic orbit (see Figure 3(a)), so the existing homoclinic orbit is no longer orbit flip for 𝑦₀=𝑀₄𝜇+h.o.t.≠0. But if 𝑦₀=0, an orbit flip homoclinic orbit could still exist, where 𝑦₀ is given by 𝐺₄=0; see Figure 3(b).

(a) 𝐹(0,𝜇)=0, 𝑦0=𝑀4𝜇+h.o.t.≠0

(b) 𝐹(0,𝜇)=0, 𝑦0=𝑀4𝜇+h.o.t.=0

Figure 3 

1-homoclinic orbit (1-H) and (1-OH).

Remark 11. There still exist some double 1-periodic orbits or triple 1-periodic orbits for the case 𝑤₄₂=0 and 𝛿_𝑢≠0; one may pursue the similar process to discuss, so we leave it here.

Acknowledgment

The project is supported by the National Natural Science Foundation of China (no. 11126097).