Constructing the Second Order Poincaré Map Based on the Hopf-Zero Unfolding Method

Abstract

We investigate the Shilnikov sense homoclinicity in a 3D system and consider the dynamical behaviors in vicinity of the principal homoclinic orbit emerging from a third order simplified system. It depends on the application of the simplest normal form theory and further evolution of the Hopf-zero singularity unfolding. For the Shilnikov sense homoclinic orbit, the complex form analytic expression is accomplished by using the power series of the manifolds surrounding the saddle-focus equilibrium. Then, the second order Poincaré map in a generally analytical style helps to portrait the double pulse dynamics existing in the tubular neighborhood of the principal homoclinic orbit.

1. Introduction

In recent years, there has been a great deal interest in understanding the dynamics and bifurcations in problems governed by 3D autonomous differential systems [1–3]. These problems include sequences of periodic doubling, existence of strange attractors, and bifurcations of the multipulse orbits. The common adapted tool for analyzing multipulse behaviors of the 3D nonlinear dynamical system is the well-known Poincaré map [4]. It starts by a standard analysis of the local and global cross-sections to the principal homoclinic orbit of the equilibrium, where the stable and unstable manifolds can intersect along the tubular neighborhood of the principal homoclinic loop, giving rise to Poincaré homoclinic structure. Among the literature, Glendinning and Sparrow [5] studied the local behavior of systems' near homoclinic orbits to the stationary equilibrium of saddle-focus type. They described the process of periodic orbit approached homoclinicity and discussed the resulting global patterns of bifurcations, such as the emergency of subsidiary homoclinic manifolds. More explicit work is offered by Gonchenko et al. [6]. They reported the research of multipulse homoclinic loops in two-parameter families of vector fields surrounding the principal homoclinic tangency and then proved the high structural instability in the neighborhood of a Shilnikov type saddle-focus.

The prosperities of the Poincaré map reflect the topological and stability of the actual flow and are basically adapted in this paper to concern the dynamics of double-pulse orbit at the tubular field. Besides that we complexly quantify the principal orbit before the analysis of the return map with the expectation to generalize the expression and obtain higher computational precise. Actually, the so-called principal homoclinicity has been frequently mentioned [7–9], but the orbit itself and its usage for the return map have not been broadly detected. Li and Zhu constructed the 𝑠 dimensional series expression of the stable and unstable manifolds [10], which formed the homoclinic orbit near the hyperbolic singular point of the Lorenz system.

With a further extension of the methodology, the Shilnikov sense principal homoclinic loop will be analytically obtained from the composition of series expression of the invariant manifolds surrounding the saddle-focus equilibrium. These quantified manifolds intersect the cross-sections locally and globally produce more explicit coordinates up to a desired order and also characterize the Poincaré homoclinic map of the system.

When talking about the complex dynamics, we acknowledge the relevance of the splitting homoclinic connection with the unfolding of the Hopf-zero singularity [11]. Meanwhile, the local dynamics presented by a normal form approach (NF) [12–15], a standard result of Birkhoff NF theory, is also applied in the initial simplification.

So, in this paper, we try to construct a second order Poincaré map based on the Hopf-zero unfolding in a common 3D system. In Section 2, it is supposed that the system exhibit the Hopf-zero singularity at the fixed parameter value 𝜇=0 of the linear part of the vector field. In Section 3, the explicit expression of the principal homoclinic loop H is given by using the invariant manifold method. In Section 4, analytical series expressions motivate our interest in chasing a more general and higher order Poincaré map from its quantified stable and unstable manifolds. The new map can be easily degenerated to the original result presented by Wiggins [16]. And it gives further facility of studying the complex dynamics, such as subsidiary or multipulse homoclinicity surrounding the principal homoclinic orbit.

2. The Shilnikov Unfolding Based on the Hopf-Zero Normal Form

Consider the following 3D system: ̇,̇,̇𝑥=𝑦+𝑓𝑥,𝑦,𝑧;𝜇𝑦=𝑧+𝑔𝑥,𝑦,𝑧;𝜇𝑧=𝜀₁𝑥+𝜀₂𝑦+𝜀₃𝑧+ℎ𝑥,𝑦,𝑧;𝜇𝑥,𝑦,𝑧,𝜇∈ℝ¹×ℝ¹×ℝ¹×ℝ¹, (1) where 𝜇 is regarded as the orbit splitting parameter and 𝑓₁,𝑓₂,𝑓₃ are 𝐶^𝑟, 𝑟≥2 and vanish along with their derivates at (𝑥,𝑦,𝑧,𝜇)=(0,0,0,0). As mentioned in [11], the unfolding of the Hopf-zero local bifurcation contains the global Shilnikov sense homoclinicity emerging from the saddle-focus equilibrium. Given that (1) enjoys Hopf-zero singularity near the origin, the normal form (NF) can be written as 𝑢̇₁=𝛾+𝑖𝜔𝑢₁+𝑎₁₁₀₁𝑢₁𝑢₂+𝑢₁^𝑚1𝑝_𝑝=1𝑗=0𝑎_{1,𝑝−𝑗+1,𝑝−𝑗,2𝑗}𝑢₁^2𝑝−𝑗𝑢₂^2𝑗+𝑢₁^𝑚2𝑝_𝑝=1𝑗=0𝑎_{1,𝑝−𝑗+1,𝑝−𝑗,2𝑗+1}𝑢₁^2𝑝−𝑗𝑢₂^2𝑗+1,𝑢̇₂=𝜆𝑢₂+^𝑚3𝑝_𝑝=1𝑗=0𝑎_{2,𝑝−𝑗,𝑝−𝑗,2𝑗}𝑢₁^2𝑝−𝑗𝑢₂^2𝑗+^𝑚1𝑝_𝑝=1𝑗=0𝑎_{2,𝑝−𝑗,𝑝−𝑗,2𝑗+1}𝑢₁^2𝑝−𝑗𝑢₂^2𝑗+1, (2) in which |𝑢₁|²=𝑢₁𝑢₁, an over-bar denotes complex conjugate; for example, 𝑢₁ denotes the complex conjugate of 𝑢₁; 𝑎_1,𝜅 and 𝑎_2,𝜅 are the NF coefficients, given in the complex and real forms, respectively. The final expressions have different forms depending on whether the total order n of the NF is an odd or even number; that is,(i)𝑚₁=𝑚₂+1=𝑚₃=(1/2)(𝑛−1), if n is odd;(ii)𝑚₁=𝑚₂=𝑚₃−1=(𝑛/2)−1, if n is even.

Here we only use the third order NF in (2) because the system is terminated in the third order [11]: 𝑢̇₁=𝛾+𝜔𝑖𝑢₁+𝑎₁₁₀₁𝑢₁𝑢₂+𝑎₁₁₀₂𝑢₁𝑢²₂+𝑎₁₂₁₀𝑢²₁𝑢₁,𝑢̇₂=𝜆𝑢₂+𝑎₂₀₀₂𝑢²₂+𝑎₂₁₁₀𝑢₁𝑢₁+𝑎₂₁₁₁𝑢₁𝑢₁𝑢₂+𝑎₂₀₀₃𝑢³₂. (3)

The dynamics of bifurcation near the structurally unstable homoclinic orbits in the Shilnikov sense requires the description on a two-parameter group (𝜇,𝜌). It takes the function as the distance between the point of the intersection of principal homoclinic H with some surface of the cross-section and the line of intersection of stable manifold 𝑊^𝑠 with the same surface of section. On the other hand, 𝜌=𝛾+𝜔𝑖/𝜆, defined in its complex form. Further discussion is restricted to the condition |𝜌_𝑟=𝛾/𝜆|<1.

Firstly, it is needed to quantify the analytical structure of the principal homoclinic orbit in the complex form at 𝜇=0. And then constructing a second order Poincaré map to prohibit a clear insight of the complex dynamics surrounding the principal orbit 𝐻 is carried out.

3. Analytical Expression of the Principal Homoclinic Orbit

The analytical expression of the principal homoclinic orbit is the combination of the 1D manifold associated with the eigenvalue 𝜆₁=𝜆 and the 2D manifold associated with 𝜆_2,3=𝛾±𝜔𝑖. So, the 1D manifold for (3) is 𝑢₁=^𝑘_𝑚=0Γ_𝑚𝑒^𝑚𝜆𝑡,𝑢₁=^𝑘_𝑚=0Γ_𝑚𝑒^𝑚𝜆𝑡,𝑢₂=^𝑘_𝑚=0Δ_𝑚𝑒^𝑚𝜆𝑡, (4) where Γ_𝑚,Γ_𝑚, and Δ_𝑚 are the coefficients to be determined. Γ_𝑚 is in the complex form and Γ_𝑚 denotes its complex conjugate. The Δ_𝑚 is a real number. Differentiate (4) with respect to time t and substitute it into (3) and then balance the same order terms of 𝑒^𝑚𝜆𝑡 on both sides of the resulting equations. That yields Γ_𝑘1=−×𝑎𝛾−𝑘𝜆+𝑖𝜔_{𝑘1101𝑚=0}Γ_𝑚Δ_𝑘−𝑚+𝑎^𝑘_1102𝑚=0^𝑘−𝑚_𝑗=0Γ_𝑚Δ_𝑗Δ_{𝑘−𝑚−𝑗}+𝑎^𝑘_1210𝑚=0^𝑘−𝑚_𝑗=0Γ_𝑚Γ_𝑗Γ_{𝑘−𝑚−𝑗},Γ_𝑘=1×−𝛾+𝑘𝜆+𝑖𝜔𝑎^𝑘_1101𝑚=0Γ_𝑚Δ_𝑘−𝑚+𝑎^𝑘_1102𝑚=0^𝑘−𝑚_𝑗=0Γ_𝑚Δ_𝑗Δ_{𝑘−𝑚−𝑗}+𝑎^𝑘_1210𝑚=0^𝑘−𝑚_𝑗=0Γ_𝑚Γ_𝑗Γ_{𝑘−𝑚−𝑗},Δ_𝑘=1()𝜆×𝑎−1+𝑘_{𝑘2002𝑚=0}Δ_𝑘−𝑚Δ_𝑚+𝑎^𝑘_2110𝑚=0Γ_𝑚Γ_𝑘−𝑚+𝑎^𝑘_2111𝑚=0^𝑘−𝑚_𝑗=0Γ_𝑚Γ_𝑗Δ_{𝑘−𝑚−𝑗}+𝑎^𝑘_2003𝑚=0^𝑘−𝑚_𝑗=0Δ_𝑚Δ_𝑗Δ_{𝑘−𝑚−𝑗}. (5)

The 2D manifolds subject to the complex enginvalues 𝜆_2,3 are 𝑢₁=𝜂+ℎ_1,2𝜂,𝜂+ℎ_1,3𝜂,𝜂+⋯+ℎ_1,𝑘𝜂,𝜂,𝑢₁=𝜂+ℎ_1,2𝜂,𝜂+ℎ_1,3𝜂,𝜂+⋯+ℎ_1,𝑘𝜂,𝜂,𝑢₂=Δ_1,0𝜂+Δ_0,1𝜂+ℎ_2,2𝜂,𝜂+ℎ_2,3𝜂,𝜂+⋯+ℎ_2,𝑘𝜂,𝜂, (6) where ℎ_1,𝑗(𝜂,𝜂)=∑^𝑗_𝑖=0Γ_{𝑖,𝑗−𝑖}𝜂^𝑖𝜂^𝑗−𝑖,ℎ_2,𝑗(𝜂,𝜂)=∑^𝑗_𝑖=0Δ_{𝑖,𝑗−𝑖}𝜂^𝑖𝜂^𝑗−𝑖,𝑗=2,…,𝑘, and Γ_𝑚,𝑛,Γ_𝑚,𝑛,Δ_𝑚,𝑛 are the coefficients to be determined, 𝑛=𝑘−𝑚. 𝜂,𝜂 can be expressed in the following complex form: 𝜂=(Γ_𝑟,1,0+𝑖Γ_𝑖,1,0)𝑒^𝜆2𝑡,𝜂=(Γ_𝑟,1,0−𝑖Γ_𝑖,1,0)𝑒^𝜆3𝑡. Differentiate (6) with respect to time 𝑡 and substitute it into (3) and then balance the same order terms of 𝑒^{(𝑚𝜆2𝑡+𝑛𝜆3𝑡)} on both sides of the resulting equation. It gives the coefficients of the 2D manifold:Γ_𝑚,𝑛=1()()𝜔𝑎−1+𝑚+𝑛𝛾+𝑖−1+𝑚−𝑛^𝑚1101_𝑚1^𝑛_=0𝑛1=0Γ_𝑚1,𝑛1Δ_{𝑚−𝑚1,𝑛−𝑛1}+𝑎^𝑚_1102𝑚1=0^𝑚−𝑚1_𝑚2^𝑛_=0𝑛1=0^𝑛−𝑛1_𝑛2=0Γ_𝑚1,𝑛1Δ_𝑚2,𝑛2Δ_{𝑚−𝑚1−𝑚2,𝑛−𝑛1−𝑛2}+𝑎^𝑚_1210𝑚1=0^𝑚−𝑚1_𝑚2^𝑛_=0𝑛1=0^𝑛−𝑛1_𝑛2=0Γ_𝑚1,𝑛1Γ_𝑚2,𝑛2Γ_{𝑛−𝑛1−𝑛2,𝑚−𝑚1−𝑚2},Γ_𝑚,𝑛=1(−1+𝑚+𝑛)𝛾−𝑖(−1+𝑚−𝑛)𝜔𝑎^𝑚_1101𝑚1^𝑛_=0𝑛1=0Γ_𝑚1,𝑛1Δ_{𝑛−𝑛1,𝑚−𝑚1}+𝑎^𝑚_1102𝑚1=0^𝑚−𝑚1_𝑚2^𝑛_=0𝑛1=0^𝑛−𝑛1_𝑛2=0Γ_𝑚1,𝑛1Δ_𝑚2,𝑛2Δ_{𝑚−𝑚1−𝑚2,𝑛−𝑛1−𝑛2}+𝑎^𝑚_1210𝑚1=0^𝑚−𝑚1_𝑚2^𝑛_=0𝑛1=0^𝑛−𝑛1_𝑛2=0Γ_𝑚1,𝑛1Γ_𝑚2,𝑛2Γ_{𝑛−𝑛1−𝑛2,𝑚−𝑚1−𝑚2},Δ_𝑚,𝑛=1()()𝜔𝑎𝑚+𝑛𝛾−𝜆+𝑖𝑚−𝑛^𝑚2002_𝑚1^𝑛_=0𝑛1=0Δ_{𝑚−𝑚1,𝑛−𝑛1}Δ_𝑚1,𝑛1+𝑎^𝑚_2110𝑚1^𝑛_=0𝑛1=0Γ_𝑚1,𝑛1Γ_{𝑛−𝑛1,𝑚−𝑚1}+𝑎^𝑚_2111𝑚1=0^𝑚−𝑚1_𝑚2^𝑛_=0𝑛1=0^𝑛−𝑛1_𝑛2=0Γ_𝑚1,𝑛1Γ_𝑚2,𝑛2Δ_{𝑚−𝑚1−𝑚2,𝑛−𝑛1−𝑛2}+𝑎^𝑚_2003𝑚1=0^𝑚−𝑚1_𝑚2^𝑛_=0𝑛1=0^𝑛−𝑛1_𝑛2=0Δ_𝑚1,𝑛1Δ_𝑚2,𝑛2Δ_{𝑚−𝑚1−𝑚2,𝑛−𝑛1−𝑛2}. (7)

These foregoing 1D and 2D invariant manifolds finally form the Shilnikov type principal homoclinic orbit H in its analytical form. It benefits the construction of the second order Poincaré map shown in Section 4.

4. The Second Order Poincaré Map

The Poincaré map is one of the most commonly used approach to study the dynamics surrounding the Shilnikov homoclinic loop including double or multipulse (circuit) orbits. In a general process, the first step to construct a return map is built from the composition of the first hit maps between the surfaces of section [4, 17]. The first hit maps are locally defined since at 𝜇=0 the surfaces of the section intersect the homoclinic cycle transversally. So, with the obtained analytical manifolds, the local linear stable and unstable manifolds are given by 𝑢₁=Γ_𝑟,1,0+𝑖Γ_𝑖,1,0𝑒^𝜆2𝑡+𝑜𝑒^𝜆2𝑡,𝑒^𝜆3𝑡,𝑢₁=Γ_𝑟,1,0−𝑖Γ_𝑖,1,0𝑒^𝜆3𝑡+𝑜𝑒^𝜆2𝑡,𝑒^𝜆3𝑡,𝑢₂=Δ₁𝑒^𝜆𝑡+𝑜𝑒^𝜆𝑡. (8)

The orbit 𝐻 intersects the complex plane (Γ_{𝑟,𝑚,𝑛},Γ_{𝑖,𝑚,𝑛}) at a countable set of points at 𝜇=0. So, we select (Γ_𝑟,1,0+𝑖Γ_𝑖,1,0,Δ₁) as the starting point of (8) in its coordinates and construct a two-dimensional cross-section Π₀ as shown in Figure 2. Following the orbit, there emerges another point 𝑢^Π1₁ belonging to the second auxiliary cross-section Π₁. Let Π₀ be a rectangle lying in the plane (Γ_{𝑟,𝑚,𝑛},Γ_{𝑖,𝑚,𝑛}) and Π₁ a rectangle parallel to the plane at 𝑢^Π1₂=Δ₁𝑒^𝜆𝑇0=𝜀. So, the Poincaré map 𝑃=𝑃₀∘𝑃₁ along the orbit should be constructed by dividing the flow close to the stationary point of its unstable manifold into two parts: a small neighborhood of the stationary point where the flow is essentially linear and a global flow which takes trajectories close to the unstable manifold away from and then back to the linear region, as shown in Figure 1.

Figure 1 

Cross-sections Π0 and Π1 near the origin.

Figure 2 

Subsidiary homoclinic orbit.

Starting from the point (Γ_𝑟,1,0+𝑖Γ_𝑖,1,0,0) on Π₀ to reach (0,𝜀) on Π₁, it produces the first local map 𝑃₀∶Π₀→Π₁. According to (8), this local map can be written as 𝑃₀∶Π₀⟶Π₁∶𝑢^Π1₁=Γ_𝑟,1,0+𝑖Γ_𝑖,1,0𝑒^𝜆2𝑇0+𝑜𝑒^𝜆2𝑇0,𝑒^𝜆3𝑇0. (9)

From the equation of 𝑢₂, the time 𝑇₀ of flight for points starting on Π₀ to reach Π₁ can be calculated from 𝑇₀1=−𝜆ΔLog¹𝜀. (10)

Substituting the flight time into map 𝑃₀, it is found that the expression for the first hit map can be expressed as 𝑃₀∶Π₀⟶Π₁∶𝑢^Π1₁=Γ_𝑟,1,0+𝑖Γ_𝑖,1,0𝑢^𝜌+𝑜𝑒^𝜆2𝑇0,𝑒^𝜆3𝑇0, (11) where 𝑢=Δ₁/𝜀,𝜌=−𝜆₂/𝜆=𝜌_𝑟+𝑖𝜌_𝑖. Since the time of flight 𝑡₀ is bounded in this progress, which generates another global map according to (6),𝑃₁∶Π₁⟶Π₀∶Γ_𝑟,1,0+𝑖Γ_𝑖,1,0−𝑥+𝑦𝑖=𝑢^Π1₁𝑒^𝜆2𝑡0+𝑓₁𝑒^𝜆2𝑡0,𝑒^𝜆3𝑡0,Δ₁−𝜇=Δ_1,0𝑢^Π1₁𝑒^𝜆2𝑡0+Δ_0,1𝑢^Π1₁𝑒^𝜆3𝑡0+𝑓₂𝑒^𝜆2𝑡0,𝑒^𝜆3𝑡0, (12) where 𝑓₁,𝑓₂ include the remaining components of 𝑒^𝜆2𝑡0,𝑒^𝜆3𝑡0, and these equations portrait the global behavior outside the equilibrium. So, the combination of the local and global map finally forms the second order Poincaré map:𝑃∶Π₀⟶Π₀∶Γ^′_𝑟,1,0+𝑖Γ^′_𝑖,1,0=𝑥+𝑦𝑖+Δ^𝜌𝑟₁𝜀^−𝜌𝑟𝑒^𝑟𝑡0Γ_𝑟,1,0+𝑖Γ_𝑖,1,0×coslnΔ^𝜌𝑖₁𝜀^−𝜌𝑖+𝜔𝑡₀+𝑖sinlnΔ^𝜌𝑖₁𝜀^−𝜌𝑖+𝜔𝑡₀+Γ_2,0Γ²_𝑟,1,0+2𝑖Γ_𝑟,1,0Γ_𝑖,1,0−Γ²_𝑖,1,0×Δ^𝜌𝑟₁𝜀^−𝜌𝑟𝑒^𝑟𝑡02cos2lnΔ^𝜌𝑖₁𝜀^−𝜌𝑖+𝜔𝑡₀+𝑖sin2lnΔ^𝜌𝑖₁𝜀^−𝜌𝑖+𝜔𝑡₀+Γ_1,1Γ²_𝑟,1,0+Γ²_𝑖,1,0Δ^𝜌𝑟₁𝜀^−𝜌𝑟𝑒^𝑟𝑡02,Δ^′₁=𝜇−2Δ^𝜌𝑟₁𝜀^−𝜌𝑟𝑒^𝑟𝑡0×Γ_𝑟,1,0Δ_𝑟,1,0−Γ_𝑖,1,0Δ_𝑖,1,0coslnΔ^𝜌𝑖₁𝜀^−𝜌𝑖+𝜔𝑡₀+Γ_𝑟,1,0Δ_𝑖,1,0+Γ_𝑖,1,0Δ_𝑟,1,0sinlnΔ^𝜌𝑖₁𝜀^−𝜌𝑖+𝜔𝑡₀−2Δ^𝜌𝑟₁𝜀^−𝜌𝑟𝑒^𝑟𝑡02×2Γ_𝑖,1,0Γ_𝑟,1,0Δ_𝑖,2,0+Γ²_𝑖,1,0−Γ²_𝑟,1,0Δ_𝑟,2,0cos2lnΔ^𝜌𝑖₁𝜀^−𝜌𝑖+𝜔𝑡₀−Γ²_𝑖,1,0−Γ²_𝑟,1,0Δ_𝑖,2,0−2Γ_𝑖,1,0Γ_𝑟,1,0Δ_𝑟,2,0sin2lnΔ^𝜌𝑖₁𝜀^−𝜌𝑖+𝜔𝑡₀+Δ_1,1Γ²_𝑖,1,0+Γ²_𝑟,1,0Δ^𝜌𝑟₁𝜀^−𝜌𝑟𝑒^𝑟𝑡02. (13)

That is a more general expression of the return map because the former results 𝑃∶Π₀⟶Π₀∶Γ^′_𝑖,1,0=1+Δ^𝜌𝑟₁Γ_𝑖,1,0coslnΔ^𝜌𝑖₁,Δ+𝜃_′1=𝜇+Δ^𝜌𝑟₁Γ_𝑖,1,0×Δ_𝑖,1,0coslnΔ^𝜌𝑖₁+𝜃−Δ_𝑟,1,0sinlnΔ^𝜌𝑖₁+𝜃, (14) as given by Gonchenko et al. and Zhou et al. [6, 7], can be easily obtained by letting 𝜀=1,𝑥=0,𝑦=1,Γ_𝑟,1,0=0,𝜔𝑡₀=𝜃,Δ^𝜌𝑟₁𝑒^𝑟𝑡0≈Δ^𝜌𝑟₁,2Δ_𝑖,1,0=Δ_𝑖,1,0,2Δ_𝑟,1,0=Δ_𝑟,1,0, and ignoring the second order components, while the new map makes itself in a broader usage because of the general expression and better computational precision.

5. Conclusions

In this paper, the second order Poincaré map in a common 3D nonlinear system is constructed. Shilnikov sense subsidiary homoclinic dynamics is also researched. Firstly, we find the strategy to analytically construct the homoclinic orbit surrounding the saddle-focus equilibrium point in terms of the complex SNF and method of invariant manifolds. By using these quantified manifolds, we can form the expression of the first and the second order Poincaré map, making their appearance in a more general style with the capacity to enhance computational precise. The further possible application of the subsidiary homoclinicity is shown in Figure 2 which is a diagram for subsidiary homoclinic orbit; the return map can portrait the dynamics unfolding from the principal orbit.

However, the analytical work lags behind the numerical results in great extent. So, to employ the higher order Poincaré map and then to unfold the successive bifurcation behaviors more simply and exactly, in a narrow parameter interval, will be the topics for further discussion.

Appendix

The SNF coefficients of the Hopf-zero bifurcation up to the third order are as follows: 𝑏_1,1,0,1=𝑎_1,1,0,1;𝑏_2,0,0,2=𝑎_2,0,0,2;𝑏_2,1,1,0=𝑎_2,1,1,0;𝑏_2,0,0,3=𝑎_2,0,0,3;𝑏_1,2,1,0𝑎=−^2,1,1,1𝑎_1,1,1,0,12𝑎_2,0,0,2−𝑎_1,1,1,0,1−𝑎_2,1,1,0𝑎_1,1,1,0,2𝑎_2,0,0,2−𝑎_1,1,1,0,1+𝑎_1,1,2,1,0−𝑎+𝑖^2,1,1,1𝑎_2,1,1,0,12𝑎_2,0,0,2−𝑎_1,1,1,0,1+𝑎_2,1,1,0𝑎_1,1,1,0,2𝑎_2,1,1,0,1𝑎_2,0,0,2𝑎_1,1,1,0,1−𝑎_2,1,1,0𝑎_1,1,1,0,2𝑎_2,1,1,0,1𝑎_2,0,0,2−𝑎_1,1,1,0,1𝑎_1,1,1,0,1−𝑎_2,1,1,0𝑎_2,1,1,0,2𝑎_2,0,0,2+𝑎_2,1,2,1,0. (A.1)

The second order coefficients of the two-dimensional manifold are as follows Γ_2,0=𝑏₁₁₀₁Δ_1,0,𝛾+𝑖𝜔Γ_2,0=𝑏₁₁₀₁Δ_0,1𝛾−𝑖𝜔,Γ_1,1=𝑏₁₁₀₁Δ_0,1,𝛾−𝑖𝜔Γ_1,1=𝑏₁₁₀₁Δ_1,0𝛾+𝑖𝜔,Γ_0,2=0,Γ_0,2Δ=0,_2,0=𝑏₂₀₀₂Δ²_1,02𝛾−𝜆+2𝑖𝜔,Δ_0,2=𝑏₂₀₀₂Δ²_0,1,Δ2𝛾−𝜆−2𝑖𝜔_1,1=𝑏₂₁₁₀+2𝑏₂₀₀₂Δ_0,1Δ_1,0.2𝛾−𝜆 (A.2)

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11272229) and Science Foundation of Tianjin education committee (Grant no. 20120902).