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

Authors: Gen Ge; Wang Wei
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 Section2, 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:(1)x˙=y+f(x,y,z;μ),y˙=z+g(x,y,z;μ),z˙=ε1x+ε2y+ε3z+h(x,y,z;μ)(x,y,z,μ)∈ℝ1×ℝ1×ℝ1×ℝ1,
where μ is regarded as the orbit splitting parameter and f1,f2,f3 are Cr, 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
(2)u˙1=(γ+iω)u1+a1101u1u2+u1∑p=1m1∑j=0pa1,p-j+1,p-j,2j(|u1|2)p-ju22j+u1∑p=1m2∑j=0pa1,p-j+1,p-j,2j+1(|u1|2)p-ju22j+1,u˙2=λu2+∑p=1m3∑j=0pa2,p-j,p-j,2j(|u1|2)p-ju22j+∑p=1m1∑j=0pa2,p-j,p-j,2j+1(|u1|2)p-ju22j+1,
in which |u1|2=u1u-1, an over-bar denotes complex conjugate; for example, u-1 denotes the complex conjugate of u1; a1,κ and a2,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), ifnis odd;(ii)
m1=m2=m3-1=(n/2)-1, ifnis even.Here we only use the third order NF in (2) because the system is terminated in the third order [11]:
(3)u˙1=(γ+ωi)u1+a1101u1u2+a1102u1u22+a1210u12u-1,u˙2=λu2+a2002u22+a2110u1u-1+a2111u1u-1u2+a2003u23.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 homoclinicH with some surface of the cross-section and the line of intersection of stable manifold Ws 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 |ρr=γ/λ|<1.Firstly, it is needed to quantify the analytical structure of the principal homoclinic orbit in the complex form atμ=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 eigenvalueλ1=λ and the 2D manifold associated with λ2,3=γ±ωi. So, the 1D manifold for (3) is
(4)u1=∑m=0kΓmemλt,u-1=∑m=0kΓ-memλt,u2=∑m=0kΔmemλt,
where Γm,Γ-m, and Δm are the coefficients to be determined. Γm is in the complex form and Γ-m denotes its complex conjugate. The Δm is a real number. Differentiate (4) with respect to timet and substitute it into (3) and then balance the same order terms of emλt on both sides of the resulting equations. That yields
(5)Γk=-1γ-kλ+iω×(a1101∑m=0kΓmΔk-m+a1102∑m=0k∑j=0k-mΓmΔjΔk-m-j+a1210∑m=0k∑j=0k-mΓmΓjΓ-k-m-j),Γ-k=1-γ+kλ+iω×(a-1101∑m=0kΓ-mΔk-m+a-1102∑m=0k∑j=0k-mΓ-mΔjΔk-m-j+a-1210∑m=0k∑j=0k-mΓ-mΓ-jΓk-m-j),Δk=1(-1+k)λ×(a2002∑m=0kΔk-mΔm+a2110∑m=0kΓmΓ-k-m+a2111∑m=0k∑j=0k-mΓmΓ-jΔk-m-j+a2003∑m=0k∑j=0k-mΔmΔjΔk-m-j).The 2D manifolds subject to the complex enginvaluesλ2,3 are
(6)u1=η+h1,2(η,η-)+h1,3(η,η-)+⋯+h1,k(η,η-),u-1=η-+h-1,2(η,η-)+h-1,3(η,η-)+⋯+h-1,k(η,η-),u2=Δ1,0η+Δ0,1η-+h2,2(η,η-)+h2,3(η,η-)+⋯+h2,k(η,η-),
where h1,j(η,η-)=∑i=0jΓi,j-iηiη-j-i,h2,j(η,η-)=∑i=0jΔi,j-iηiη-j-i,j=2,…,k, and Γm,n,Γ-m,n,Δm,n are the coefficients to be determined, n=k-m. η,η- can be expressed in the following complex form: η=(Γr,1,0+iΓi,1,0)eλ2t,η-=(Γr,1,0-iΓi,1,0)eλ3t. 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:(7)Γm,n=1(-1+m+n)γ+i(-1+m-n)ω(a1101∑m1=0m∑n1=0nΓm1,n1Δm-m1,n-n1mmmmmmmmmmmmmmmmmmmmm+a1102∑m1=0m∑m2=0m-m1∑n1=0n∑n2=0n-n1Γm1,n1Δm2,n2Δm-m1-m2,n-n1-n2mmmmmmmmmmmmmmmmmmmmm+a1210∑m1=0m∑m2=0m-m1∑n1=0n∑n2=0n-n1Γm1,n1Γm2,n2Γ-n-n1-n2,m-m1-m2),Γ-m,n=1(-1+m+n)γ-i(-1+m-n)ω(a-1101∑m1=0m∑n1=0nΓ-m1,n1Δn-n1,m-m1mmmmmmmmmmmmmmmmmmmmmmm+a-1102∑m1=0m∑m2=0m-m1∑n1=0n∑n2=0n-n1Γ-m1,n1Δm2,n2Δm-m1-m2,n-n1-n2mmmmmmmmmmmmmmmmmmmmmmm+a-1210∑m1=0m∑m2=0m-m1∑n1=0n∑n2=0n-n1Γ-m1,n1Γ-m2,n2Γn-n1-n2,m-m1-m2),Δm,n=1(m+n)γ-λ+i(m-n)ω(a2002∑m1=0m∑n1=0nΔm-m1,n-n1Δm1,n1+a2110∑m1=0m∑n1=0nΓm1,n1Γ-n-n1,m-m1mmmmmmmmmmmmmmmmmmm+a2111∑m1=0m∑m2=0m-m1∑n1=0n∑n2=0n-n1Γm1,n1Γ-m2,n2Δm-m1-m2,n-n1-n2mmmmmmmmmmmmmmmmmmm+a2003∑m1=0m∑m2=0m-m1∑n1=0n∑n2=0n-n1Δm1,n1Δm2,n2Δm-m1-m2,n-n1-n2).These foregoing 1D and 2D invariant manifolds finally form the Shilnikov type principal homoclinic orbitH 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
(8)u1=(Γr,1,0+iΓi,1,0)eλ2t+o(eλ2t,eλ3t),u-1=(Γr,1,0-iΓi,1,0)eλ3t+o(eλ2t,eλ3t),u2=Δ1eλt+o(eλt).The orbitH intersects the complex plane (Γr,m,n,Γi,m,n) at a countable set of points at μ=0. So, we select (Γr,1,0+iΓi,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 u1Π1 belonging to the second auxiliary cross-section Π1. Let Π0 be a rectangle lying in the plane (Γr,m,n,Γi,m,n) and Π1 a rectangle parallel to the plane at u2Π1=Δ1eλT0=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-sectionsΠ0 and Π1 near the origin.Figure 2
Subsidiary homoclinic orbit.Starting from the point(Γr,1,0+iΓi,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
(9)P0:Π0⟶Π1:u1Π1=(Γr,1,0+iΓi,1,0)eλ2T0+o(eλ2T0,eλ3T0).From the equation ofu2, the time T0 of flight for points starting on Π0 to reach Π1 can be calculated from
(10)T0=-1λLog(Δ1ε).Substituting the flight time into mapP0, it is found that the expression for the first hit map can be expressed as
(11)P0:Π0⟶Π1:u1Π1=(Γr,1,0+iΓi,1,0)uρ+o(eλ2T0,eλ3T0),
where u=Δ1/ε,ρ=-λ2/λ=ρr+iρi. Since the time of flight t0 is bounded in this progress, which generates another global map according to (6),(12)P1:Π1⟶Π0:{Γr,1,0+iΓi,1,0-(x+yi)=u1Π1eλ2t0+f1(eλ2t0,eλ3t0),Δ1-μ=Δ1,0u1Π1eλ2t0+Δ0,1u-1Π1eλ3t0+f2(eλ2t0,eλ3t0),
where f1,f2 include the remaining components of eλ2t0,eλ3t0, 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:(13)P:Π0⟶Π0:{Γr,1,0′+iΓi,1,0′=(x+yi)+Δ1ρrε-ρrert0(Γr,1,0+iΓi,1,0)×{cos[ln(Δ1ρiε-ρi)+ωt0]+isin[ln(Δ1ρiε-ρi)+ωt0]}+Γ2,0(Γr,1,02+2iΓr,1,0Γi,1,0-Γi,1,02)×(Δ1ρrε-ρrert0)2{cos2[ln(Δ1ρiε-ρi)+ωt0]+isin2[ln(Δ1ρiε-ρi)+ωt0]}+Γ1,1(Γr,1,02+Γi,1,02)(Δ1ρrε-ρrert0)2,Δ1′=μ-2Δ1ρrε-ρrert0×{(Γr,1,0Δr,1,0-Γi,1,0Δi,1,0)cos[ln(Δ1ρiε-ρi)+ωt0]+(Γr,1,0Δi,1,0+Γi,1,0Δr,1,0)sin[ln(Δ1ρiε-ρi)+ωt0]}-2(Δ1ρrε-ρrert0)2×{[2Γi,1,0Γr,1,0Δi,2,0+(Γi,1,02-Γr,1,02)Δr,2,0]cos2[ln(Δ1ρiε-ρi)+ωt0]-[(Γi,1,02-Γr,1,02)Δi,2,0-2Γi,1,0Γr,1,0Δr,2,0]sin2[ln(Δ1ρiε-ρi)+ωt0]}+Δ1,1(Γi,1,02+Γr,1,02)(Δ1ρrε-ρrert0)2.That is a more general expression of the return map because the former results(14)P:Π0⟶Π0:{Γi,1,0′=1+Δ1ρrΓi,1,0cos(lnΔ1ρi+θ),Δ1′=μ+Δ1ρrΓi,1,0×[Δi,1,0cos(lnΔ1ρi+θ)-Δr,1,0sin(lnΔ1ρi+θ)],
as given by  Gonchenko et al. and Zhou et al. [6, 7], can be easily obtained by letting ε=1,x=0,y=1,Γr,1,0=0,ωt0=θ,Δ1ρrert0≈Δ1ρr,2Δi,1,0=Δi,1,0,2Δr,1,0=Δr,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 Figure2 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.



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*Source: 294162-2013-10-10.xml*