# Dynamical Analysis and Simulation of a New Lorenz-Like Chaotic System

Authors: You Li; Ming Zhao; Fengjie Geng
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This work presents and investigates a new chaotic system with eight terms. By numerical simulation, the two-scroll chaotic attractor is found for some certain parameters. And, by theoretical analysis, we discuss the dynamical behavior of the new-type Lorenz-like chaotic system. Firstly, the local dynamical properties, such as the distribution and the local stability of all equilibrium points, the local stable and unstable manifolds, and the Hopf bifurcations, are all revealed as the parameters varying in the space of parameters. Secondly, by applying the way of Poincaré compactification inℝ3, the dynamics at infinity are clearly analyzed. Thirdly, combining the dynamics at finity and those at infinity, the global dynamical behaviors are formulated. Especially, we have proved the existence of the infinite heteroclinic orbits. Furthermore, all obtained theoretical results in this paper are further verified by numerical simulations.


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## Body

## 1. Introduction

Chaos, as a magical and charming nonlinear phenomenon, has attracted attentions of many scholars in nonlinear dynamics community. This is not only because it is a mysterious and profound subject but also because it is beneficial to many practical applications. Especially, with the development of computer science, chaos has been applied to secure communication [1], biomedical system analysis [2], power system protection [3], and fluid mixing [4]. Accordingly, due to these influential applications, the synchronization and control of chaotic systems have fascinated numerous research studies; one may see [5–8] and references therein.Ever since Lorenz discovered the chaotic phenomena in a simple 3D nonlinear ODE in 1963, more and more specialists devoted themselves to chaotic dynamics. Some of them were interested in finding different kinds of new chaotic models and discussed their dynamical properties, for example, Lü system [9], Chen system [10], Lorenz-type system [11–14], and other new chaotic system [15–19], while some of them focused their attentions on revisiting the existing chaotic system and explored some new phenomena which had not been found before [20–28]. All of these research studies are beneficial to reveal the essence of chaos.As we know, if we add or change the linear or nonlinear term of existing chaotic system, then a new type chaotic system will be achieved. In [29], the authors constructed a new Lorenz-like chaotic system:(1)x˙=ay−x,y˙=bx−lxz,z˙=−cz+hx2+ky2.They pointed out that when the parameters satisfya,b,l,c,h,k=10,40,1,2.5,2,2, this new system has a butterfly-shaped attractor.Based on Lü system, Li and Ou [11] presented and considered a new chaotic system:(2)x˙=ay−x,y˙=dy−xz,z˙=−bz+fx2+gxy,where a>0, b,d∈ℝ, g≥0, f≥0, and g+f>0. The authors showed that when the parameters satisfy a=10, d=6, b=3, f=1, and g=0, system (2) will have a chaotic attractor. The local dynamical behavior, the existence for homoclinic orbits, and the existence for heteroclinic orbits of this new system were established.After that, the authors [6] proposed the following new 3D chaotic system with a closed circle of the equilibrium points:(3)x˙=z,y˙=−zx2+ay−by2−cxz,z˙=y−0.5x2+0.15x2−1,where a,b,c>0. They studied the dynamics of system (3), and they designed an electronic circuit enforcement of this new system to illustrate its feasibility.Recently, Wang and Li [30] replaced x with x2 in a Lorenz-type system and obtained a new nonlinear system as(4)x˙=ay−x,y˙=cx2−x2z,z˙=−bz+x2y,where a,b,c∈ℝ and a≠0. Although the chaotic attractor has not been found so far, they proved that system (4) has a single heteroclinic trajectory, which is the new character of the proposed system.Motivated by the results about the different new nonlinear systems, in this paper, by adding a linear term in system (2), we propose a new system:(5)x˙=ay−x,y˙=−dx+cy−xz,z˙=−bz+fx2+gxy,where a>0, b,c,d∈ℝ, g≥0, f≥0, and g+f>0. Notice that there are eight terms in this system, and it is obvious that it is not topologically equivalent to any form of the generalized Lorenz system. Furthermore, Lü system [9] and system (2) are included in our new system.By takinga,d,c,b,f,g=10,2,6,3,2,1,1 and t=5000, we can achieve the Lyapunov exponents of system (5) as λLE1=0.2603, λLE2=−0.0008, and λLE3=−7.2595, which means that there is a chaotic phenomena in system (5). And, the numerical simulation in Figure 1 shows that system (5) has a two-scroll chaotic attractor which looks like Lorenz attractor. Hence, this new system can be called as Lorenz-like system. Due to the above discovery, we are inspired to explore some more and new complex dynamical behaviors of system (5).Figure 1
(a-b) Phase portrait and Lyapunov exponents for system (5) when a,d,c,b,f,g=10,2,6,3,2,1,1.
(a)(b)Therefore, in this paper, our main purpose is to explore the global dynamics of system (5) in detail. It is worthy pointing out that the dynamics at infinity especially the existence of infinite heteroclinic orbits have not been mentioned in [11].This paper is constructed as follows. In Section2, the basic dynamics of system (5), including the existence and their local properties of the equilibrium points, are provided. Particularly, the specific expressions of local stable (resp. unstable) manifolds and the corresponding numerical simulations are also presented. In Section 3, by virtue of the Poincaré compactification in ℝ3, we present the dynamical behaviors on the sphere at infinity, where we find the infinite heteroclinic orbits for some certain parameters. In Section 4, we give a brief conclusion and some future work.


## 2. Local Dynamical Behavior of System (5)

In the following, we consider the local dynamical properties for system (5), including the existence and the stability of all equilibrium points, Hopf bifurcation, and the local structure of trajectories.First, we indicate that system (5) satisfies the following properties:(1)
Symmetry and invariance.(2)
Normality.(3)
Dissipativity.These can be easily proved with similar arguments to [11], so we omit it here.


### 2.1. The Existence of Equilibrium Points

In order to discuss the existence of equilibrium points for system (5), we need to solve the algebraic equations as follows:(6)ay−x=0,−dx+cy−xz=0,−bz+fx2+gxy=0.By some simple computations, we achieve the following results.Theorem 1.
For system (5), we have the following true statements:(1)
Ifbc−d<0 or c−d=0, then there exists exactly one equilibrium point E0.(2)
Ifb=0, then there exist nonisolated equilibrium points Ez for z∈R.(3)
Ifbc−d>0, then there exist three equilibrium points E0 and E±.
Here,E0=0,0,0, Ez=0,0,z, and E±=±bc−d/g+f,±bc−d/g+f,c−d.


### 2.2. Local Dynamical Properties of the Equilibrium PointE0

For the equilibrium pointE0 of system (5), we can get its Jacobian matrix A0:(7)A0=−aa0−dc000−b.Then, the characteristic equation ofA0 is obtained:(8)pλ=λ+bλ2+a−cλ−ac−d=0.We will discuss the local dynamical properties of equilibrium pointE0 in the following three sections.


#### 2.2.1. The Casec−d>0

Due to equation (8), we obtain three eigenvalues:(9)λ1,2=−a−c±c−a2+4ac−d2,λ3=−b.Their corresponding eigenvectors are(10)v1,2=−a+c±Δ−2d,1,0,v3=0,0,1,where Δ=a−c2+4ac−d. It is not difficult to see that λ1,2 are both real roots that satisfy λ1>0 and λ2<0. The root λ3>0 (resp. λ3<0) for b<0 (resp. b>0).Applying the center manifold theorem, we can derive the stability ofE0.Lemma 1.
Forc−d>0 and b<0, E0 is a saddle. And, it has a 2D unstable manifold that embraces z-axis and a 1D stable manifold Wlocs, which can be expressed by(11)Wlocs=xyz|x≪1,y=a+c−Δ2ax+Ox2,z=2afb+c−a+Δ+ga+c−Δb+c−a+Δ2abb−2a+2c−8a2c−dx2+Ox3..
The corresponding illustrative phase portraits are shown in Figure2.Figure 2
(a–d) Phase portraits for system (5) when a,b,c,d,f,g=3,−1,2,0,1,0: (a) in the xy-plane; (b) in the xz-plane; (c) in the yz-plane; (d) in R3.
(a)(b)(c)(d)Proof.
Whenc−d>0 and b<0, it is obvious that λ1>0, λ3>0, and λ2<0. This means that E0 is a saddle, which consists of a 2D unstable manifold that includes z-axis and a 1D stable manifold Wlocs, which can be expressed by(12)Wlocs=xyz|x≪1y=Hxz=KxA11H′0=λ21H′0,H0=K0=0,where A1=−aa−dc.
Lety=Hx=h1x+h2x2+Ox3 and z=Kx=k1x+k2x2+Ox3. Put them into system (5) and then compare the coefficients of the same term on both sides, and we achieve(13)ah1−a−ch1+d=0,3ah1−2a−ch2=−k1,ah1−1k1=−bk1,ak1h2+2k2h1−1=−bk2+f+gh1.
It follows from the matrix equation,(14)−aa−dc1H′0=λ21H′0,that h1=a+c−Δ/2a. So, we achieve(15)k1=0,k2=2afb+c−a+Δ+gc+a−Δb+c−a+Δ2abb−2a+2c−8a2c−d,h2=0.
The proof of Lemma1 is then finished.Lemma 2.
Forc−d>0 and b>0, E0 is a saddle. And, it has a 2D stable manifold that embraces z-axis and a 1D unstable manifold Wlocu, which is expressed as(16)Wlocu=xyzx≪1,y=a+c+Δ2ax+Ox2,z=2afb+c−a+Δ+ga+c+Δb+c−a−Δ2abb−2a+2c−8a2c−dx2+Ox3.Proof.
Whenc−d>0 and b>0, it is obvious that λ2<0, λ3<0, and λ1>0. Therefore, E0 is a saddle, which consists of a 2D stable manifold that includes z-axis, and a 1D unstable manifold Wlocu, which is written as(17)Wlocu=xyz|x≪1y=Hxz=KxA21H′0=λ11H′0,H0=K0=0,where A2=−aa−dc.
Lety=Hx=h1x+h2x2+Ox3 and z=Kx=k1x+k2x2+Ox3. Similar to Lemma 1, we may derive that(18)h1=a+c+Δ2a,h2=0,k1=0,k2=2afb+c−a+Δ+gc+a+Δb+c−a−Δ2abb−2a+2c−8a2c−d.
Thus, we complete the proof of Lemma2.


#### 2.2.2. The Casec−d=0

Lemma 3.
Supposec−d=0, then we have the following true statements:(1)
Ifd−a<0, then(i)
Forb>0, E0 is a nonhyperbolic point and asymptotically stable. The corresponding illustrative phase portrait is depicted in Figure 3.(ii)
Forb<0, E0 is a nonhyperbolic point, and it is of saddle-center type.(2)
Ifd−a>0, then(i)
Forb>0, E0 is a nonhyperbolic point, and it is of saddle-center type.(ii)
Forb<0, E0 is a nonhyperbolic point and unstable.Figure 3
Phase portrait for system (5) when the parameters satisfy a,b,c,d,f,g=3,1,2,2,1,0 in R3. Obviously, E0 is asymptotically stable.Proof.
(1)
Whenc−d=0, d<a, and b>0, E0 has three eigenvalues λ1<0, λ3<0, and λ2=0. Thus, it is nonhyperbolic. In the following, we will use the center manifold theory to further determine its stability.Let(19)xyz=1−a01−d0001x1y1z1.Then, system (5) turns into(20)x˙1y˙1z˙1=0000−a−d000−bx1y1z1+−aa−dx1−ay1z1−1a−dx1−ay1z1fx1−ay12+gx1−ay1x1−dy1.Based on the center manifold theory, we can determine the stability of the equilibrium pointE0 by investigating a first-order ODE restrict to its center manifold, and the center manifold can be expressed as(21)Wlocc=x1,y1,z1∈R3|y1=h1x1,z1=h2x1,x1<δ,h10=h20=h1′0=h2′0=0,withδ sufficiently small.Assumeh1x1=α1x12+β1x13+Ox4 and h2x1=α2x12+β2x13+Ox4; then, it must satisfy(22)aa−dh1′x1x1−ah1x1h2x1−a−dh1x1−1a−dx1−ah1x1h2x1=0,aa−dh2′x1x1−ah1x1h2x1−bh2x1+fx1−ah1x12+gx1−ah1x1x1−dh1x1=0.By some computations, we obtain(23)h1x1=−f+gba−d2x13+Ox14,h2x1=f+gbx12+Ox14.Hence, system (5) restricted to the center manifold may be written as(24)x˙1=−af+gba−dx13+Ox14.It then follows that the equilibrium pointE0 is asymptotically stable.Forc−d=0, d<a, and b<0, E0 has three eigenvalues λ1<0, λ3>0, and λ2=0, which implies E0 is nonhyperbolic point and is of saddle-center type. Hence, we finish the proof of Lemma 3 (1).(2)
Whenc−d=0, d>a, and b>0, the equilibrium point E0 has three eigenvalues λ1>0, λ3<0, and λ2=0, which means E0 is nonhyperbolic point and is of saddle-center type.
Whenc−d=0, d>a, and b<0, the equilibrium point E0 has three eigenvalues λ1>0, λ3>0, and λ2=0. So, E0 is nonhyperbolic point and unstable. Hence, we finish the proof of Lemma 3 (2).Remark 1.
Ifd=a and c−d=0, system (5) will have a center manifold with high-order terms.


#### 2.2.3. The Casec−d<0

Whenc−d<0, the dynamical properties of the equilibrium point E0 are formulated in Table 1.Table 1
The properties of equilibrium pointE0 for c−d<0.
ConditionsEigenvaluesTypec−a>0b>0Reλ1,2>0, λ3<0Saddle-focusb<0Reλ1,2>0, λ3>0Unstable node-focusc−a=0b>0λ1,2=±2ad−ci, λ3<0Non-hyperbolicb<0λ1,2=±2ad−ci, λ3>0Non-hyperbolicc−a<0b>0Reλ1,2<0, λ3<0Stable node-focusb<0Reλ1,2<0, λ3>0Saddle-focusEspecially, whenc−d<0, b>0, and c−a=0, we have the lemma as follows.Lemma 4.
Supposec−d<0, c−a=0, and b>0, then the Hopf bifurcation will happen to system (5) at the equilibrium point E0, and there exists an orbit with the period T=π/ad−c. The corresponding illustrative phase portrait is shown in Figure 4.Figure 4
Phase portrait for system (5) when a,b,c,d,f,g=3,1,3,4,1,0.Proof.
Whenc−d<0, b>0, and c−a=0, the characteristic equation (8) has roots λ3=−b<0 and λ1,2=±2ad−ci. Then, we can obtain(25)dReλ1dc|c=a=12>0.
By Hopf bifurcation theory [31], a Hopf bifurcation will happen to system (5) at E0. This means a limit cycle will occur at c=a, which has initial period T=π/ad−c and increases with c. So, the proof of Lemma 4 is then completed.
Summing up the above discussions, we get the main theorem as follows.Theorem 2.
Assumea>0, g≥0, f≥0, g+f>0, and b,c,d∈ℝ, then the equilibrium point E0 of system (5) has the local dynamical properties which are given in Table 2.Table 2
Local dynamical properties of equilibrium pointE0.
ConditionsPropertyc−dd−ac−ab>0>0A 2DWlocs and a 1D Wlocu<0A 1DWlocs and a 2D Wlocu=0>0>0A 1DWlocs, a 1D Wlocu, and a 1D Wlocc<0A 2DWlocu and a 1D Wlocc=0>0A 1DWlocs and a 2D Wlocc<0A 1DWlocu and a 2D Wlocc<0>0A 2DWlocs and a 1D Wlocc<0A 1DWlocs, a 1D Wlocu, and a 1D Wlocc<0>0>0A 1DWlocs and a 2D Wlocu<0A 3DWlocu=0>0A 1DWlocs and a 2D Wlocc<0A 1DWlocu and a 2D Wlocc<0>0A 3DWlocs<0A 2DWlocs and a 1D Wlocu


### 2.3. Local Dynamic Behavior of the Nonisolated Equilibrium PointsEz

The characteristic equation at the nonisolated pointsEz of system (5) is(26)λλ2+a−cλ−ac−d−z=0.By analyzing the characteristic roots of the above equation, we can get the results as follows.Theorem 3.
Suppose thata>0, b=0, c,d∈ℝ, g≥0, f≥0, and g+f>0, then the equilibrium points Ez have the local dynamical properties which are summed in Table 3.Table 3
Local dynamical properties of the nonisolated equilibrium pointsEz.
ConditionsPropertyc−a>0z>c−dA1DWlocs, a 1DWlocu, and a 1DWloccz=c−dA1DWlocu and a 2DWloccz<c−dA2DWlocu and a 1DWloccc−a=0z>c−dMay be fold-Hopf bifurcationz=c−dA3DWloccz<c−dA1DWlocs, a 1DWlocu, and a 1DWloccc−a<0z>c−dA2DWlocs and a 1DWloccz=c−dA1DWlocs and a 2DWloccz<c−dA1DWlocs, a 1DWlocu, and a 1DWlocc


### 2.4. Local Dynamical Behavior of the Equilibrium PointsE±

In what follows, we deal with the dynamical behavior ofE±, which means bc−d>0. By symmetry of (5), we know that the stability of E− can be achieved from that of the equilibrium point E+. Thus, we just consider the local dynamical behavior of E+.For convenience in our later discussion, we first give some results of the polynomial with real coefficient:(27)pλ=λ3+a1λ2+a2λ+a3.Set three roots of equationpλ=0 be λ1, λ2, and λ3. Then, it is easy to obtain(28)λ1+λ2+λ3=−a1,λ1λ2+λ2λ3+λ1λ3=a2,λ1λ2λ3=−a3.Let(29)H=a1a301a200a1a3.We can obtain the principal minors forH which are H1=a1, H2=a1a2−a3, and H3=a3H2. Thus, we have the following results.Lemma 5.
Supposea3>0, then we have the following true statements:(1)
pλ=0 has no root λi=0i=1,2,3.(2)
Ifpλ=0 has a real root and a pair of pure imaginary roots, then H2=0.(3)
If one ofHi satisfies Hi<0, then pλ=0 has at least a root λi with Reλi>0i=1,2,3.Proof.
(1)
It follows froma3=−λ1λ2λ3>0 that λi≠0i=1,2,3. So, we finish the proof of (1).(2)
Whena3>0, suppose that pλ=0 has a real root λ1 and a pair of pure imaginary roots λ2,3=±ωiω>0. It follows from (1) that λ1≠0; then, from (11), we can derive H2=a1a2−a3=0, completing the proof of (2).(3)
Assumea3>0, when pλ=0 has three real roots or a pair of conjugate roots with nonzero real parts and a real root; then, it follows from the Routh–Hurwitz criterion that pλ=0 has at least a root λi with Reλi>0i=1,2,3. And, when pλ=0 has a real root λ1 and a pair of pure imaginary roots λ2,3, if H2<0, then according to (2), we know H2=0, a contradiction; if H3<0, then from H3=a3H2 and a3>0, we can get H2<0, this is also a contradiction; if H1<0, from the first equation of (12), we obtain λ1=λ1+λ2+λ3=−H1>0. So, if one of Hii=1,2,3 satisfies Hi<0, then pλ=0 has at least a root λi with positive real part. Thus, (3) is proved.
The Jacobian matrixA at E+ of system (5) is(30)A=−aa0−cc−Λ2f+gΛgΛ−b,where Λ=bc−d/f+g. And, its characteristic equation is(31)kλ=λ3+a+b−cλ2+ba−dg+cff+gλ+2abc−d=0.
Denote three characteristic roots of (31) by λ1, λ2, and λ3. Combining equations (27) and (31), we obtain(32)λ1+λ2+λ3=−a+b−c,λ1λ2+λ2λ3+λ1λ3=ba−dg+cff+g,λ1λ2λ3=2abc−d.
Then,(33)H=a+b−c2abc−d01ba−dg+cff+g00a+b−c2abc−d,and the principal minors for H are(34)H1=a+b−c,H2=ba−dg+cff+ga+b−c−2ac−d,H3=2abc−dH2.
LetH2=bpc/f+g, where(35)pc=fc2−3ag+f−dg+a+bfc+a+bag+f−dg+2adf+g.
Forpc, we have the following properties:(1)
Iff=0 and d≤a, then pc=0 has a root c0, which can be expressed as(36)c0=a2+ab+d−bd3a−d.(2)
Iff>0, it follows that(i)
Ford<a, pc=0 has two real roots c±, which can be expressed in this form:(37)c±=3ag+f−dg+a+bf±ag+f+a+bf+dg2+8aa−df+g22f.(ii)
Ford=a and 2ag+f+bf≠0, pc=0 has two real roots c±, which can be expressed as(38)c±=3ag+f−dg+a+bf±ag+f+a+bf+dg2+8aa−df+g22f.(iii)
Ford=a and 2af+g+bf=0, pc=0 has a unique real root c¯=a. Based on the above analysis, it is not difficult to deduce the following lemmas about c0 and c±.Lemma 6.
Whenb>0 and d<a, it follows(1)
Iff=0, then d<c0<a+b.(2)
Iff>0, then d<c−<a+b<c+.Lemma 7.
Whenb<0 and d<a, then we have the following true statements:(1)
Forf=0,(i)
Ifa+b<d<a, then a+b<c0<d.(ii)
Ifa+b=d<a, then a+b=c0=d.(iii)
Ifd<a+b<a, then d<c0<a+b.(2)
Forf>0,(i)
Ifa+b<d<a, then a+b<c−<d<c+.(ii)
Ifa+b=d<a, then a+b=c−=d<c+.(iii)
Ifd<a+b<a, then d<c−<a+b<c+.
Then, we can conclude the results in the following.Theorem 4.
Supposeb>0 and d<a, then we have the following true statements:(1)
Forf=0,(i)
Ifd<c<c0, then E+ is asymptotically stable.(ii)
Ifc>c0, then E+ is unstable.(iii)
Ifc=c0, then system (5) will undergo a Hopf bifurcation at E+ and a periodic orbit with period T=2π/2abc0−d/a+b−c0 will occur. The illustrative phase portraits are depicted in Figure 5.(2)
Forf>0,(i)
Ifd<c<c−, then E+ is asymptotical stable.(ii)
Ifc>c−, then E+ is unstable.(iii)
Ifc=c−, then system (5) undergoes a Hopf bifurcation at E+ and an orbit with periodic T=2π/2abc−−d/a+b−c− will happen. The corresponding illustrative phase portraits are shown in Figure 6.Proof.
Based on Routh–Hurwitz criterion, we know the necessary and sufficient condition forλ1, λ2, and λ3 to have negative real parts is H1, H2, H3>0. When b>0, it follows from the existence of E+ that c>d. Then, due to (34), we achieve(39)H1=a+b−c⇔c<a+b,H2=ba−dg+cff+ga+b−c−2ac−d⇔pc>0,H3=2abc−dH2⇔H2>0.
Forf=0, based on Lemma 6 (1), we know that d<c0<a+b holds. Hence, for d<c<c0, we have H1, H2, H3>0, which shows E+ is asymptotical stable, while, for c>c0, it is easy to derive H2<0; therefore, Lemma 5 (3) implies that has at least a root λi with Reλi>0i∈1,2,3, so E+ is unstable. Thus, we complete the proofs of (1) (i) and (1) (ii).
Forf>0, the proofs of (2) (i) and (2) (ii) are similar to that of (1) (i) and (1) (ii), so we omitted them here.
Next, we prove (2) (iii). For f>0 and c=c−, we know that kλ has a pair of pure imaginary roots λ1,2=±2abc−−d/a+b−c−i=±ωi and one negative real root λ3=c−−a+b. Complicated calculations yield(40)dReλ1dc|c=c−=bfa+b−c−+f+gω+2ab2f+gω+a+b−c−2>0.
According to the Hopf bifurcation theory [31], one knows that a limit cycle occurs at c=c−, which increases with respect to c and has the initial period T=2π/2abc−−d/a+b−c−. The above arguments imply that a Hopf bifurcation will happen at E+ of system (5). Using the same method as (2) (iii), we can prove the results of (1) (iii).Figure 5
Phase portraits for system (5) when a,b,d,f,g=0.8,1,0.4,0,1 and (a) c=c0=0.69, (b) c=0.6<c0, and (c) c=0.8>c0.
(a)(b)(c)Figure 6
Phase portraits for system (5) when a,b,d,f,g=1,1,−1,1,0 and (a) c=c−=0, (b) c=−0.5<c−, and (c) c=0.5>c−.
(a)(b)(c)Theorem 5.
Supposeb<0 and d<a, then E+ is unstable. The corresponding illustrative phase portraits are depicted in Figure 7.Figure 7
Phase portraits for system (5) when a>d and (a) a,b,c,d,f,g=3,−1,1,2,1,0 and (b) a,b,c,d,f,g=3,−1,1,2,0,1.
(a)(b)Proof.
Whenb<0, in order to guarantee the existence of E+, we require c<d. If f=0, based on the statements of Lemma 7 (1), we know that there exists at least a Hi<0i=1,2,3; then, according to Lemma 5 (3), we get that kλ=0 has at least one root with positive real part; thus, E+ is unstable. If f>0, with similar argument to the proof of f=0, combining Lemma 7 (2) and Lemma 5 (3), we achieve the unstability of E+. Consequently, we finish the proof of Theorem 5.Theorem 6.
Assumed=a, then E+ is unstable. The corresponding illustrative phase portraits are depicted in Figure 8.Figure 8
Phase portraits for system (5) when a=d and (a) a,b,c,d,f,g=3,−1,2,3,1,0, (b) a,b,c,d,f,g=3,−1,2,3,0,1, (c) a,b,c,d,f,g=3,1,4,3,1,0, and (d) a,b,c,d,f,g=3,1,4,3,0,1.
(a)(b)(c)(d)Proof.
Ifd=a, let us consider b>0 and b<0, respectively, as follows:Case 1 (b>0): it follows from the existence of E+ that c>a holds. If f=0, then c0=a, which means c>c0=a; thus, H2<0. If f>0, then pc=0 has two real roots, which leads to a=c−<a+b<c+; thus, when a<c≤a+b, we have H2<0, while, for c>a+b, we may obtain H1<0. Hence, according to the results of Lemma 5 (3), we know that kλ has at least one root λii∈1,2,3 with Reλi>0. So, E+ is unstable.Case 2 (b<0): the condition c<a is provided by the existence of E+. If f=0, then c0=a, which leads to c<c0=a; thus, H2<0. If f>0 and 2af+g+bf=0, then pc=0 has one real root c¯ and c<c¯=a, which shows H2<0. If f>0 and 2af+g+bf>0, then pc=0 has two real roots and a+b<c−, so, when c≥c−, we have H1<0, while, for c<c−, H2<0 is fulfilled. If f>0 and 2af+g+bf<0, then pc=0 has two real roots and a+b<c+; hence, for c≥c+, H1<0 holds, while, for c<c+, H2<0 is true. Therefore, based on the conclusion of Lemma 5 (3), we get that kλ=0 has at least one root λii∈1,2,3 with Reλi>0. Hence, the equilibrium point E+ is unstable, which finishes Theorem 6.
BecauseE+ and E− are symmetrical, the following results can be achieved.Theorem 7.
Whena>0, b,c,d∈ℝ, d≤a, g≥0, f≥0, and g+f>0, the local dynamical properties of the equilibrium points E± are summed up in Table 4.Table 4
Local dynamical properties ofE±.
bd−afcDynamical behavior ofE±>0<0=0c<c0Asymptotical stablec=c0Hopf bifurcationc>c0Unstable>0c<c−Asymptotical stablec=c−Hopf bifurcationc>c−Unstable=0Unstable<0≤0UnstableRemark 2.
For the cased>a, we cannot find an effective method to classify the parameters to investigate the dynamical behavior of E±, but we present numerical simulation for parameters a,b,c,d,f,g=2,1,4,3,0,1 and a,b,c,d,f,g=2,−1,3,4,0,1 which are shown in Figure 9.Figure 9
Phase portraits for system (5) when the parameters satisfy (a) a,b,c,d,f,g=2,1,4,3,0,1 and (b) a,b,c,d,f,g=2,−1,3,4,0,1.
(a)(b)


## 2.1. The Existence of Equilibrium Points

In order to discuss the existence of equilibrium points for system (5), we need to solve the algebraic equations as follows:(6)ay−x=0,−dx+cy−xz=0,−bz+fx2+gxy=0.By some simple computations, we achieve the following results.Theorem 1.
For system (5), we have the following true statements:(1)
Ifbc−d<0 or c−d=0, then there exists exactly one equilibrium point E0.(2)
Ifb=0, then there exist nonisolated equilibrium points Ez for z∈R.(3)
Ifbc−d>0, then there exist three equilibrium points E0 and E±.
Here,E0=0,0,0, Ez=0,0,z, and E±=±bc−d/g+f,±bc−d/g+f,c−d.


## 2.2. Local Dynamical Properties of the Equilibrium PointE0

For the equilibrium pointE0 of system (5), we can get its Jacobian matrix A0:(7)A0=−aa0−dc000−b.Then, the characteristic equation ofA0 is obtained:(8)pλ=λ+bλ2+a−cλ−ac−d=0.We will discuss the local dynamical properties of equilibrium pointE0 in the following three sections.


### 2.2.1. The Casec−d>0

Due to equation (8), we obtain three eigenvalues:(9)λ1,2=−a−c±c−a2+4ac−d2,λ3=−b.Their corresponding eigenvectors are(10)v1,2=−a+c±Δ−2d,1,0,v3=0,0,1,where Δ=a−c2+4ac−d. It is not difficult to see that λ1,2 are both real roots that satisfy λ1>0 and λ2<0. The root λ3>0 (resp. λ3<0) for b<0 (resp. b>0).Applying the center manifold theorem, we can derive the stability ofE0.Lemma 1.
Forc−d>0 and b<0, E0 is a saddle. And, it has a 2D unstable manifold that embraces z-axis and a 1D stable manifold Wlocs, which can be expressed by(11)Wlocs=xyz|x≪1,y=a+c−Δ2ax+Ox2,z=2afb+c−a+Δ+ga+c−Δb+c−a+Δ2abb−2a+2c−8a2c−dx2+Ox3..
The corresponding illustrative phase portraits are shown in Figure2.Figure 2
(a–d) Phase portraits for system (5) when a,b,c,d,f,g=3,−1,2,0,1,0: (a) in the xy-plane; (b) in the xz-plane; (c) in the yz-plane; (d) in R3.
(a)(b)(c)(d)Proof.
Whenc−d>0 and b<0, it is obvious that λ1>0, λ3>0, and λ2<0. This means that E0 is a saddle, which consists of a 2D unstable manifold that includes z-axis and a 1D stable manifold Wlocs, which can be expressed by(12)Wlocs=xyz|x≪1y=Hxz=KxA11H′0=λ21H′0,H0=K0=0,where A1=−aa−dc.
Lety=Hx=h1x+h2x2+Ox3 and z=Kx=k1x+k2x2+Ox3. Put them into system (5) and then compare the coefficients of the same term on both sides, and we achieve(13)ah1−a−ch1+d=0,3ah1−2a−ch2=−k1,ah1−1k1=−bk1,ak1h2+2k2h1−1=−bk2+f+gh1.
It follows from the matrix equation,(14)−aa−dc1H′0=λ21H′0,that h1=a+c−Δ/2a. So, we achieve(15)k1=0,k2=2afb+c−a+Δ+gc+a−Δb+c−a+Δ2abb−2a+2c−8a2c−d,h2=0.
The proof of Lemma1 is then finished.Lemma 2.
Forc−d>0 and b>0, E0 is a saddle. And, it has a 2D stable manifold that embraces z-axis and a 1D unstable manifold Wlocu, which is expressed as(16)Wlocu=xyzx≪1,y=a+c+Δ2ax+Ox2,z=2afb+c−a+Δ+ga+c+Δb+c−a−Δ2abb−2a+2c−8a2c−dx2+Ox3.Proof.
Whenc−d>0 and b>0, it is obvious that λ2<0, λ3<0, and λ1>0. Therefore, E0 is a saddle, which consists of a 2D stable manifold that includes z-axis, and a 1D unstable manifold Wlocu, which is written as(17)Wlocu=xyz|x≪1y=Hxz=KxA21H′0=λ11H′0,H0=K0=0,where A2=−aa−dc.
Lety=Hx=h1x+h2x2+Ox3 and z=Kx=k1x+k2x2+Ox3. Similar to Lemma 1, we may derive that(18)h1=a+c+Δ2a,h2=0,k1=0,k2=2afb+c−a+Δ+gc+a+Δb+c−a−Δ2abb−2a+2c−8a2c−d.
Thus, we complete the proof of Lemma2.


### 2.2.2. The Casec−d=0

Lemma 3.
Supposec−d=0, then we have the following true statements:(1)
Ifd−a<0, then(i)
Forb>0, E0 is a nonhyperbolic point and asymptotically stable. The corresponding illustrative phase portrait is depicted in Figure 3.(ii)
Forb<0, E0 is a nonhyperbolic point, and it is of saddle-center type.(2)
Ifd−a>0, then(i)
Forb>0, E0 is a nonhyperbolic point, and it is of saddle-center type.(ii)
Forb<0, E0 is a nonhyperbolic point and unstable.Figure 3
Phase portrait for system (5) when the parameters satisfy a,b,c,d,f,g=3,1,2,2,1,0 in R3. Obviously, E0 is asymptotically stable.Proof.
(1)
Whenc−d=0, d<a, and b>0, E0 has three eigenvalues λ1<0, λ3<0, and λ2=0. Thus, it is nonhyperbolic. In the following, we will use the center manifold theory to further determine its stability.Let(19)xyz=1−a01−d0001x1y1z1.Then, system (5) turns into(20)x˙1y˙1z˙1=0000−a−d000−bx1y1z1+−aa−dx1−ay1z1−1a−dx1−ay1z1fx1−ay12+gx1−ay1x1−dy1.Based on the center manifold theory, we can determine the stability of the equilibrium pointE0 by investigating a first-order ODE restrict to its center manifold, and the center manifold can be expressed as(21)Wlocc=x1,y1,z1∈R3|y1=h1x1,z1=h2x1,x1<δ,h10=h20=h1′0=h2′0=0,withδ sufficiently small.Assumeh1x1=α1x12+β1x13+Ox4 and h2x1=α2x12+β2x13+Ox4; then, it must satisfy(22)aa−dh1′x1x1−ah1x1h2x1−a−dh1x1−1a−dx1−ah1x1h2x1=0,aa−dh2′x1x1−ah1x1h2x1−bh2x1+fx1−ah1x12+gx1−ah1x1x1−dh1x1=0.By some computations, we obtain(23)h1x1=−f+gba−d2x13+Ox14,h2x1=f+gbx12+Ox14.Hence, system (5) restricted to the center manifold may be written as(24)x˙1=−af+gba−dx13+Ox14.It then follows that the equilibrium pointE0 is asymptotically stable.Forc−d=0, d<a, and b<0, E0 has three eigenvalues λ1<0, λ3>0, and λ2=0, which implies E0 is nonhyperbolic point and is of saddle-center type. Hence, we finish the proof of Lemma 3 (1).(2)
Whenc−d=0, d>a, and b>0, the equilibrium point E0 has three eigenvalues λ1>0, λ3<0, and λ2=0, which means E0 is nonhyperbolic point and is of saddle-center type.
Whenc−d=0, d>a, and b<0, the equilibrium point E0 has three eigenvalues λ1>0, λ3>0, and λ2=0. So, E0 is nonhyperbolic point and unstable. Hence, we finish the proof of Lemma 3 (2).Remark 1.
Ifd=a and c−d=0, system (5) will have a center manifold with high-order terms.


### 2.2.3. The Casec−d<0

Whenc−d<0, the dynamical properties of the equilibrium point E0 are formulated in Table 1.Table 1
The properties of equilibrium pointE0 for c−d<0.
ConditionsEigenvaluesTypec−a>0b>0Reλ1,2>0, λ3<0Saddle-focusb<0Reλ1,2>0, λ3>0Unstable node-focusc−a=0b>0λ1,2=±2ad−ci, λ3<0Non-hyperbolicb<0λ1,2=±2ad−ci, λ3>0Non-hyperbolicc−a<0b>0Reλ1,2<0, λ3<0Stable node-focusb<0Reλ1,2<0, λ3>0Saddle-focusEspecially, whenc−d<0, b>0, and c−a=0, we have the lemma as follows.Lemma 4.
Supposec−d<0, c−a=0, and b>0, then the Hopf bifurcation will happen to system (5) at the equilibrium point E0, and there exists an orbit with the period T=π/ad−c. The corresponding illustrative phase portrait is shown in Figure 4.Figure 4
Phase portrait for system (5) when a,b,c,d,f,g=3,1,3,4,1,0.Proof.
Whenc−d<0, b>0, and c−a=0, the characteristic equation (8) has roots λ3=−b<0 and λ1,2=±2ad−ci. Then, we can obtain(25)dReλ1dc|c=a=12>0.
By Hopf bifurcation theory [31], a Hopf bifurcation will happen to system (5) at E0. This means a limit cycle will occur at c=a, which has initial period T=π/ad−c and increases with c. So, the proof of Lemma 4 is then completed.
Summing up the above discussions, we get the main theorem as follows.Theorem 2.
Assumea>0, g≥0, f≥0, g+f>0, and b,c,d∈ℝ, then the equilibrium point E0 of system (5) has the local dynamical properties which are given in Table 2.Table 2
Local dynamical properties of equilibrium pointE0.
ConditionsPropertyc−dd−ac−ab>0>0A 2DWlocs and a 1D Wlocu<0A 1DWlocs and a 2D Wlocu=0>0>0A 1DWlocs, a 1D Wlocu, and a 1D Wlocc<0A 2DWlocu and a 1D Wlocc=0>0A 1DWlocs and a 2D Wlocc<0A 1DWlocu and a 2D Wlocc<0>0A 2DWlocs and a 1D Wlocc<0A 1DWlocs, a 1D Wlocu, and a 1D Wlocc<0>0>0A 1DWlocs and a 2D Wlocu<0A 3DWlocu=0>0A 1DWlocs and a 2D Wlocc<0A 1DWlocu and a 2D Wlocc<0>0A 3DWlocs<0A 2DWlocs and a 1D Wlocu


## 2.2.1. The Casec−d>0

Due to equation (8), we obtain three eigenvalues:(9)λ1,2=−a−c±c−a2+4ac−d2,λ3=−b.Their corresponding eigenvectors are(10)v1,2=−a+c±Δ−2d,1,0,v3=0,0,1,where Δ=a−c2+4ac−d. It is not difficult to see that λ1,2 are both real roots that satisfy λ1>0 and λ2<0. The root λ3>0 (resp. λ3<0) for b<0 (resp. b>0).Applying the center manifold theorem, we can derive the stability ofE0.Lemma 1.
Forc−d>0 and b<0, E0 is a saddle. And, it has a 2D unstable manifold that embraces z-axis and a 1D stable manifold Wlocs, which can be expressed by(11)Wlocs=xyz|x≪1,y=a+c−Δ2ax+Ox2,z=2afb+c−a+Δ+ga+c−Δb+c−a+Δ2abb−2a+2c−8a2c−dx2+Ox3..
The corresponding illustrative phase portraits are shown in Figure2.Figure 2
(a–d) Phase portraits for system (5) when a,b,c,d,f,g=3,−1,2,0,1,0: (a) in the xy-plane; (b) in the xz-plane; (c) in the yz-plane; (d) in R3.
(a)(b)(c)(d)Proof.
Whenc−d>0 and b<0, it is obvious that λ1>0, λ3>0, and λ2<0. This means that E0 is a saddle, which consists of a 2D unstable manifold that includes z-axis and a 1D stable manifold Wlocs, which can be expressed by(12)Wlocs=xyz|x≪1y=Hxz=KxA11H′0=λ21H′0,H0=K0=0,where A1=−aa−dc.
Lety=Hx=h1x+h2x2+Ox3 and z=Kx=k1x+k2x2+Ox3. Put them into system (5) and then compare the coefficients of the same term on both sides, and we achieve(13)ah1−a−ch1+d=0,3ah1−2a−ch2=−k1,ah1−1k1=−bk1,ak1h2+2k2h1−1=−bk2+f+gh1.
It follows from the matrix equation,(14)−aa−dc1H′0=λ21H′0,that h1=a+c−Δ/2a. So, we achieve(15)k1=0,k2=2afb+c−a+Δ+gc+a−Δb+c−a+Δ2abb−2a+2c−8a2c−d,h2=0.
The proof of Lemma1 is then finished.Lemma 2.
Forc−d>0 and b>0, E0 is a saddle. And, it has a 2D stable manifold that embraces z-axis and a 1D unstable manifold Wlocu, which is expressed as(16)Wlocu=xyzx≪1,y=a+c+Δ2ax+Ox2,z=2afb+c−a+Δ+ga+c+Δb+c−a−Δ2abb−2a+2c−8a2c−dx2+Ox3.Proof.
Whenc−d>0 and b>0, it is obvious that λ2<0, λ3<0, and λ1>0. Therefore, E0 is a saddle, which consists of a 2D stable manifold that includes z-axis, and a 1D unstable manifold Wlocu, which is written as(17)Wlocu=xyz|x≪1y=Hxz=KxA21H′0=λ11H′0,H0=K0=0,where A2=−aa−dc.
Lety=Hx=h1x+h2x2+Ox3 and z=Kx=k1x+k2x2+Ox3. Similar to Lemma 1, we may derive that(18)h1=a+c+Δ2a,h2=0,k1=0,k2=2afb+c−a+Δ+gc+a+Δb+c−a−Δ2abb−2a+2c−8a2c−d.
Thus, we complete the proof of Lemma2.


## 2.2.2. The Casec−d=0

Lemma 3.
Supposec−d=0, then we have the following true statements:(1)
Ifd−a<0, then(i)
Forb>0, E0 is a nonhyperbolic point and asymptotically stable. The corresponding illustrative phase portrait is depicted in Figure 3.(ii)
Forb<0, E0 is a nonhyperbolic point, and it is of saddle-center type.(2)
Ifd−a>0, then(i)
Forb>0, E0 is a nonhyperbolic point, and it is of saddle-center type.(ii)
Forb<0, E0 is a nonhyperbolic point and unstable.Figure 3
Phase portrait for system (5) when the parameters satisfy a,b,c,d,f,g=3,1,2,2,1,0 in R3. Obviously, E0 is asymptotically stable.Proof.
(1)
Whenc−d=0, d<a, and b>0, E0 has three eigenvalues λ1<0, λ3<0, and λ2=0. Thus, it is nonhyperbolic. In the following, we will use the center manifold theory to further determine its stability.Let(19)xyz=1−a01−d0001x1y1z1.Then, system (5) turns into(20)x˙1y˙1z˙1=0000−a−d000−bx1y1z1+−aa−dx1−ay1z1−1a−dx1−ay1z1fx1−ay12+gx1−ay1x1−dy1.Based on the center manifold theory, we can determine the stability of the equilibrium pointE0 by investigating a first-order ODE restrict to its center manifold, and the center manifold can be expressed as(21)Wlocc=x1,y1,z1∈R3|y1=h1x1,z1=h2x1,x1<δ,h10=h20=h1′0=h2′0=0,withδ sufficiently small.Assumeh1x1=α1x12+β1x13+Ox4 and h2x1=α2x12+β2x13+Ox4; then, it must satisfy(22)aa−dh1′x1x1−ah1x1h2x1−a−dh1x1−1a−dx1−ah1x1h2x1=0,aa−dh2′x1x1−ah1x1h2x1−bh2x1+fx1−ah1x12+gx1−ah1x1x1−dh1x1=0.By some computations, we obtain(23)h1x1=−f+gba−d2x13+Ox14,h2x1=f+gbx12+Ox14.Hence, system (5) restricted to the center manifold may be written as(24)x˙1=−af+gba−dx13+Ox14.It then follows that the equilibrium pointE0 is asymptotically stable.Forc−d=0, d<a, and b<0, E0 has three eigenvalues λ1<0, λ3>0, and λ2=0, which implies E0 is nonhyperbolic point and is of saddle-center type. Hence, we finish the proof of Lemma 3 (1).(2)
Whenc−d=0, d>a, and b>0, the equilibrium point E0 has three eigenvalues λ1>0, λ3<0, and λ2=0, which means E0 is nonhyperbolic point and is of saddle-center type.
Whenc−d=0, d>a, and b<0, the equilibrium point E0 has three eigenvalues λ1>0, λ3>0, and λ2=0. So, E0 is nonhyperbolic point and unstable. Hence, we finish the proof of Lemma 3 (2).Remark 1.
Ifd=a and c−d=0, system (5) will have a center manifold with high-order terms.


## 2.2.3. The Casec−d<0

Whenc−d<0, the dynamical properties of the equilibrium point E0 are formulated in Table 1.Table 1
The properties of equilibrium pointE0 for c−d<0.
ConditionsEigenvaluesTypec−a>0b>0Reλ1,2>0, λ3<0Saddle-focusb<0Reλ1,2>0, λ3>0Unstable node-focusc−a=0b>0λ1,2=±2ad−ci, λ3<0Non-hyperbolicb<0λ1,2=±2ad−ci, λ3>0Non-hyperbolicc−a<0b>0Reλ1,2<0, λ3<0Stable node-focusb<0Reλ1,2<0, λ3>0Saddle-focusEspecially, whenc−d<0, b>0, and c−a=0, we have the lemma as follows.Lemma 4.
Supposec−d<0, c−a=0, and b>0, then the Hopf bifurcation will happen to system (5) at the equilibrium point E0, and there exists an orbit with the period T=π/ad−c. The corresponding illustrative phase portrait is shown in Figure 4.Figure 4
Phase portrait for system (5) when a,b,c,d,f,g=3,1,3,4,1,0.Proof.
Whenc−d<0, b>0, and c−a=0, the characteristic equation (8) has roots λ3=−b<0 and λ1,2=±2ad−ci. Then, we can obtain(25)dReλ1dc|c=a=12>0.
By Hopf bifurcation theory [31], a Hopf bifurcation will happen to system (5) at E0. This means a limit cycle will occur at c=a, which has initial period T=π/ad−c and increases with c. So, the proof of Lemma 4 is then completed.
Summing up the above discussions, we get the main theorem as follows.Theorem 2.
Assumea>0, g≥0, f≥0, g+f>0, and b,c,d∈ℝ, then the equilibrium point E0 of system (5) has the local dynamical properties which are given in Table 2.Table 2
Local dynamical properties of equilibrium pointE0.
ConditionsPropertyc−dd−ac−ab>0>0A 2DWlocs and a 1D Wlocu<0A 1DWlocs and a 2D Wlocu=0>0>0A 1DWlocs, a 1D Wlocu, and a 1D Wlocc<0A 2DWlocu and a 1D Wlocc=0>0A 1DWlocs and a 2D Wlocc<0A 1DWlocu and a 2D Wlocc<0>0A 2DWlocs and a 1D Wlocc<0A 1DWlocs, a 1D Wlocu, and a 1D Wlocc<0>0>0A 1DWlocs and a 2D Wlocu<0A 3DWlocu=0>0A 1DWlocs and a 2D Wlocc<0A 1DWlocu and a 2D Wlocc<0>0A 3DWlocs<0A 2DWlocs and a 1D Wlocu


## 2.3. Local Dynamic Behavior of the Nonisolated Equilibrium PointsEz

The characteristic equation at the nonisolated pointsEz of system (5) is(26)λλ2+a−cλ−ac−d−z=0.By analyzing the characteristic roots of the above equation, we can get the results as follows.Theorem 3.
Suppose thata>0, b=0, c,d∈ℝ, g≥0, f≥0, and g+f>0, then the equilibrium points Ez have the local dynamical properties which are summed in Table 3.Table 3
Local dynamical properties of the nonisolated equilibrium pointsEz.
ConditionsPropertyc−a>0z>c−dA1DWlocs, a 1DWlocu, and a 1DWloccz=c−dA1DWlocu and a 2DWloccz<c−dA2DWlocu and a 1DWloccc−a=0z>c−dMay be fold-Hopf bifurcationz=c−dA3DWloccz<c−dA1DWlocs, a 1DWlocu, and a 1DWloccc−a<0z>c−dA2DWlocs and a 1DWloccz=c−dA1DWlocs and a 2DWloccz<c−dA1DWlocs, a 1DWlocu, and a 1DWlocc


## 2.4. Local Dynamical Behavior of the Equilibrium PointsE±

In what follows, we deal with the dynamical behavior ofE±, which means bc−d>0. By symmetry of (5), we know that the stability of E− can be achieved from that of the equilibrium point E+. Thus, we just consider the local dynamical behavior of E+.For convenience in our later discussion, we first give some results of the polynomial with real coefficient:(27)pλ=λ3+a1λ2+a2λ+a3.Set three roots of equationpλ=0 be λ1, λ2, and λ3. Then, it is easy to obtain(28)λ1+λ2+λ3=−a1,λ1λ2+λ2λ3+λ1λ3=a2,λ1λ2λ3=−a3.Let(29)H=a1a301a200a1a3.We can obtain the principal minors forH which are H1=a1, H2=a1a2−a3, and H3=a3H2. Thus, we have the following results.Lemma 5.
Supposea3>0, then we have the following true statements:(1)
pλ=0 has no root λi=0i=1,2,3.(2)
Ifpλ=0 has a real root and a pair of pure imaginary roots, then H2=0.(3)
If one ofHi satisfies Hi<0, then pλ=0 has at least a root λi with Reλi>0i=1,2,3.Proof.
(1)
It follows froma3=−λ1λ2λ3>0 that λi≠0i=1,2,3. So, we finish the proof of (1).(2)
Whena3>0, suppose that pλ=0 has a real root λ1 and a pair of pure imaginary roots λ2,3=±ωiω>0. It follows from (1) that λ1≠0; then, from (11), we can derive H2=a1a2−a3=0, completing the proof of (2).(3)
Assumea3>0, when pλ=0 has three real roots or a pair of conjugate roots with nonzero real parts and a real root; then, it follows from the Routh–Hurwitz criterion that pλ=0 has at least a root λi with Reλi>0i=1,2,3. And, when pλ=0 has a real root λ1 and a pair of pure imaginary roots λ2,3, if H2<0, then according to (2), we know H2=0, a contradiction; if H3<0, then from H3=a3H2 and a3>0, we can get H2<0, this is also a contradiction; if H1<0, from the first equation of (12), we obtain λ1=λ1+λ2+λ3=−H1>0. So, if one of Hii=1,2,3 satisfies Hi<0, then pλ=0 has at least a root λi with positive real part. Thus, (3) is proved.
The Jacobian matrixA at E+ of system (5) is(30)A=−aa0−cc−Λ2f+gΛgΛ−b,where Λ=bc−d/f+g. And, its characteristic equation is(31)kλ=λ3+a+b−cλ2+ba−dg+cff+gλ+2abc−d=0.
Denote three characteristic roots of (31) by λ1, λ2, and λ3. Combining equations (27) and (31), we obtain(32)λ1+λ2+λ3=−a+b−c,λ1λ2+λ2λ3+λ1λ3=ba−dg+cff+g,λ1λ2λ3=2abc−d.
Then,(33)H=a+b−c2abc−d01ba−dg+cff+g00a+b−c2abc−d,and the principal minors for H are(34)H1=a+b−c,H2=ba−dg+cff+ga+b−c−2ac−d,H3=2abc−dH2.
LetH2=bpc/f+g, where(35)pc=fc2−3ag+f−dg+a+bfc+a+bag+f−dg+2adf+g.
Forpc, we have the following properties:(1)
Iff=0 and d≤a, then pc=0 has a root c0, which can be expressed as(36)c0=a2+ab+d−bd3a−d.(2)
Iff>0, it follows that(i)
Ford<a, pc=0 has two real roots c±, which can be expressed in this form:(37)c±=3ag+f−dg+a+bf±ag+f+a+bf+dg2+8aa−df+g22f.(ii)
Ford=a and 2ag+f+bf≠0, pc=0 has two real roots c±, which can be expressed as(38)c±=3ag+f−dg+a+bf±ag+f+a+bf+dg2+8aa−df+g22f.(iii)
Ford=a and 2af+g+bf=0, pc=0 has a unique real root c¯=a. Based on the above analysis, it is not difficult to deduce the following lemmas about c0 and c±.Lemma 6.
Whenb>0 and d<a, it follows(1)
Iff=0, then d<c0<a+b.(2)
Iff>0, then d<c−<a+b<c+.Lemma 7.
Whenb<0 and d<a, then we have the following true statements:(1)
Forf=0,(i)
Ifa+b<d<a, then a+b<c0<d.(ii)
Ifa+b=d<a, then a+b=c0=d.(iii)
Ifd<a+b<a, then d<c0<a+b.(2)
Forf>0,(i)
Ifa+b<d<a, then a+b<c−<d<c+.(ii)
Ifa+b=d<a, then a+b=c−=d<c+.(iii)
Ifd<a+b<a, then d<c−<a+b<c+.
Then, we can conclude the results in the following.Theorem 4.
Supposeb>0 and d<a, then we have the following true statements:(1)
Forf=0,(i)
Ifd<c<c0, then E+ is asymptotically stable.(ii)
Ifc>c0, then E+ is unstable.(iii)
Ifc=c0, then system (5) will undergo a Hopf bifurcation at E+ and a periodic orbit with period T=2π/2abc0−d/a+b−c0 will occur. The illustrative phase portraits are depicted in Figure 5.(2)
Forf>0,(i)
Ifd<c<c−, then E+ is asymptotical stable.(ii)
Ifc>c−, then E+ is unstable.(iii)
Ifc=c−, then system (5) undergoes a Hopf bifurcation at E+ and an orbit with periodic T=2π/2abc−−d/a+b−c− will happen. The corresponding illustrative phase portraits are shown in Figure 6.Proof.
Based on Routh–Hurwitz criterion, we know the necessary and sufficient condition forλ1, λ2, and λ3 to have negative real parts is H1, H2, H3>0. When b>0, it follows from the existence of E+ that c>d. Then, due to (34), we achieve(39)H1=a+b−c⇔c<a+b,H2=ba−dg+cff+ga+b−c−2ac−d⇔pc>0,H3=2abc−dH2⇔H2>0.
Forf=0, based on Lemma 6 (1), we know that d<c0<a+b holds. Hence, for d<c<c0, we have H1, H2, H3>0, which shows E+ is asymptotical stable, while, for c>c0, it is easy to derive H2<0; therefore, Lemma 5 (3) implies that has at least a root λi with Reλi>0i∈1,2,3, so E+ is unstable. Thus, we complete the proofs of (1) (i) and (1) (ii).
Forf>0, the proofs of (2) (i) and (2) (ii) are similar to that of (1) (i) and (1) (ii), so we omitted them here.
Next, we prove (2) (iii). For f>0 and c=c−, we know that kλ has a pair of pure imaginary roots λ1,2=±2abc−−d/a+b−c−i=±ωi and one negative real root λ3=c−−a+b. Complicated calculations yield(40)dReλ1dc|c=c−=bfa+b−c−+f+gω+2ab2f+gω+a+b−c−2>0.
According to the Hopf bifurcation theory [31], one knows that a limit cycle occurs at c=c−, which increases with respect to c and has the initial period T=2π/2abc−−d/a+b−c−. The above arguments imply that a Hopf bifurcation will happen at E+ of system (5). Using the same method as (2) (iii), we can prove the results of (1) (iii).Figure 5
Phase portraits for system (5) when a,b,d,f,g=0.8,1,0.4,0,1 and (a) c=c0=0.69, (b) c=0.6<c0, and (c) c=0.8>c0.
(a)(b)(c)Figure 6
Phase portraits for system (5) when a,b,d,f,g=1,1,−1,1,0 and (a) c=c−=0, (b) c=−0.5<c−, and (c) c=0.5>c−.
(a)(b)(c)Theorem 5.
Supposeb<0 and d<a, then E+ is unstable. The corresponding illustrative phase portraits are depicted in Figure 7.Figure 7
Phase portraits for system (5) when a>d and (a) a,b,c,d,f,g=3,−1,1,2,1,0 and (b) a,b,c,d,f,g=3,−1,1,2,0,1.
(a)(b)Proof.
Whenb<0, in order to guarantee the existence of E+, we require c<d. If f=0, based on the statements of Lemma 7 (1), we know that there exists at least a Hi<0i=1,2,3; then, according to Lemma 5 (3), we get that kλ=0 has at least one root with positive real part; thus, E+ is unstable. If f>0, with similar argument to the proof of f=0, combining Lemma 7 (2) and Lemma 5 (3), we achieve the unstability of E+. Consequently, we finish the proof of Theorem 5.Theorem 6.
Assumed=a, then E+ is unstable. The corresponding illustrative phase portraits are depicted in Figure 8.Figure 8
Phase portraits for system (5) when a=d and (a) a,b,c,d,f,g=3,−1,2,3,1,0, (b) a,b,c,d,f,g=3,−1,2,3,0,1, (c) a,b,c,d,f,g=3,1,4,3,1,0, and (d) a,b,c,d,f,g=3,1,4,3,0,1.
(a)(b)(c)(d)Proof.
Ifd=a, let us consider b>0 and b<0, respectively, as follows:Case 1 (b>0): it follows from the existence of E+ that c>a holds. If f=0, then c0=a, which means c>c0=a; thus, H2<0. If f>0, then pc=0 has two real roots, which leads to a=c−<a+b<c+; thus, when a<c≤a+b, we have H2<0, while, for c>a+b, we may obtain H1<0. Hence, according to the results of Lemma 5 (3), we know that kλ has at least one root λii∈1,2,3 with Reλi>0. So, E+ is unstable.Case 2 (b<0): the condition c<a is provided by the existence of E+. If f=0, then c0=a, which leads to c<c0=a; thus, H2<0. If f>0 and 2af+g+bf=0, then pc=0 has one real root c¯ and c<c¯=a, which shows H2<0. If f>0 and 2af+g+bf>0, then pc=0 has two real roots and a+b<c−, so, when c≥c−, we have H1<0, while, for c<c−, H2<0 is fulfilled. If f>0 and 2af+g+bf<0, then pc=0 has two real roots and a+b<c+; hence, for c≥c+, H1<0 holds, while, for c<c+, H2<0 is true. Therefore, based on the conclusion of Lemma 5 (3), we get that kλ=0 has at least one root λii∈1,2,3 with Reλi>0. Hence, the equilibrium point E+ is unstable, which finishes Theorem 6.
BecauseE+ and E− are symmetrical, the following results can be achieved.Theorem 7.
Whena>0, b,c,d∈ℝ, d≤a, g≥0, f≥0, and g+f>0, the local dynamical properties of the equilibrium points E± are summed up in Table 4.Table 4
Local dynamical properties ofE±.
bd−afcDynamical behavior ofE±>0<0=0c<c0Asymptotical stablec=c0Hopf bifurcationc>c0Unstable>0c<c−Asymptotical stablec=c−Hopf bifurcationc>c−Unstable=0Unstable<0≤0UnstableRemark 2.
For the cased>a, we cannot find an effective method to classify the parameters to investigate the dynamical behavior of E±, but we present numerical simulation for parameters a,b,c,d,f,g=2,1,4,3,0,1 and a,b,c,d,f,g=2,−1,3,4,0,1 which are shown in Figure 9.Figure 9
Phase portraits for system (5) when the parameters satisfy (a) a,b,c,d,f,g=2,1,4,3,0,1 and (b) a,b,c,d,f,g=2,−1,3,4,0,1.
(a)(b)


## 3. Dynamics at Infinity on System (5)

In this section, the dynamics at infinity on system (5) will be discussed with the aid of the way to Poincaré compactification in ℝ3 [32, 33].


### 3.1. The Infinite Equilibrium Points of System (5)

To study the dynamics at infinity, we should investigate the Poincaré compactification for system (5) in the local charts Ui and Vii=1,2,3, respectively.


#### 3.1.1. In the Local ChartsU1 and V1

Based on [32, 33], the Poincaré compactification pX for system (5) in the chart U1 may be displayed as(41)z˙1=−z2−dz3+a+cz1z3−az12z3,z˙2=f+gz1+a−bz2z3−az1z2z3,z˙3=az32−az1z32.As we know, to study the points of the sphereS2 at infinity, one needs to set z3=0; it then follows that(42)z˙1=−z2,z˙2=f+gz1.One may get that system (42) has no equilibrium point for g=0, while, for g>0, system (42) has a unique equilibrium point −f/g,0. That is to say, system (41) has no equilibrium point for g=0 and has a single equilibrium point P1−f/g,0,0 for g>0. Moreover, a first integral,(43)Hz1,z2=z1+fg2+z22g,has been found in system (42), which shows the equilibrium point P1 of system (41) is a center. And, the phase portrait for (43) has been drawn in Figure 10, which helps us to get the corresponding phase portrait for our system (5) on the sphere at infinity.Figure 10
The phase portrait ofHz1,z2=z1+f/g2+z22/g when f,g=1,1.In order to find out the dynamics of the local property ofP1, we can analyze the flow on the center manifold related to its eigenvalue 0 and achieve the statement listed as follows.Lemma 8.
Wheng>0, the equilibrium point P1 of system (40) has a 1D center manifold Wlocc, and it is unstable along Wlocc.Proof.
Since the equilibrium pointP1 has eigenvalues λ1,2=±gi and λ3=0, which implies P1 has a 1D Wlocc, set k1=z1+f/g, k2=z2, and k3=z3, which translates the singular point P1 into the origin O0,0,0. Then, system (41) is converted into(44)k˙1=−k2−dk3+a+ck1−f/gk3−ak1−f/g2k3,k˙2=gk1+a−bk2k3−ak1−f/gk2k3,k˙3=ak32−ak1−f/gk32.
Thus, the stability ofP1 restricted on the one-dimensional center manifold can be established according to that of the origin on a center manifold. Assume that the graph of a function h:ℝ⟶ℝ2 is presented with k1,k2=hk3=h1k3,h2k3, fulfilling(45)h0=0,0,Dh0=0,−m.
Expandh at k3=0 with Taylor series, which gives(46)h1k3=∑i=2∞aik3i,h2k3=−mk3+∑i=2∞aik3i.
Substituting them into (44) and comparing the coefficients of k3 on both sides, we obtain(47)h1k3=−mbgk32+Ok34,h2k3=a−cmbgk33+Ok34.
Then, we derivek˙3=a1+f/gk32+Ok34, which implies that when z3>0, P1 is local unstable along its center manifold.
Notice that multiplying -1 in the compactified vector fieldpX in U1 can give the compactified vector field pX in V1, so the flow in V1 is the same as the flow in U1 if we reverse the time. We have to point out that z3<0 should be considered in the neighborhood of infinity in V1, see Figure 11.Figure 11
(a) Phase portrait for system (40) when a,b,c,d,f,g=1,1,1,1,1,1. (b) Phase portrait of system (42) with f,g=1,1.
(a)(b)


#### 3.1.2. In the Local ChartsU2 and V2

The Poincaré compactificationpX for system (5) in U2 has the following form:(48)z˙1=az3−a+cz1z3+dz12z3+z12z22,z˙2=−b+cz2z3+dz1z2z3+gz1+fz12+z1z22,z˙3=−cz32+z1z2z3+dz1z32.By settingz3=0, system (48) becomes(49)z˙1=z12z2,z˙2=gz1+fz12+z1z22.By some simple computations, one knows that, forg=0 and f>0 or g>0 and f=0, system (49) has a line of equilibrium points 0,z2, while, for g>0 and f>0, system (48) has an equilibrium point −g/f,0 except the line of equilibrium points 0,z2, which means system (48) has a line of equilibria Q10,z2,0 for g=0 and f>0 or g>0 and f=0, while, for g>0 and f>0, system (48) has a line of the equilibrium points Q10,z2,0 and an isolated equilibrium point Q2−g/f,0,0.Moreover, system (49) has a first integral,(50)Hz1,z2=z22+2fz1+gz12,which helps us to get the phase portrait of system (51), one may see Figure 12.Figure 12
The phase portrait for system (49) when f,g=0,1.Similar to the situation ofU1 and V1, we can get the flow of system (5) in V2.


#### 3.1.3. In the Local ChartsU3 and V3

Similarly, one may has the Poincaré compactificationpX for system (5) in the local chart U3 listed as(51)z˙1=b−az1z3+az2z3−fz13−gz12z2,z˙2=−dz1z3+b+cz2z3−z1−fz12z2−gz1z22,z˙3=bz32−gz1z2z3−fz12z3,which turns into the following form with z3=0:(52)z˙1=−fz13−gz12z2,z˙2=−z1−fz12z2−gz1z22.It is not difficult to see that there is a first integral in system (52):(53)Hz1,z2=gz22+2fz1z2z12,and then, the phase portrait of system (52) will be achieved, which is shown in Figure 13.Figure 13
The phase portrait for system (27) when f,g=1,1.With similar arguments above, we know that system (51) has a line of the equilibrium points R10,z2,0. Moreover, the dynamics in the local charts V3 may be easily obtained.Summarizing the analysis provided above, one may achieve the following result about the infinite dynamics of system (5) on the sphere.Theorem 8.
Supposeg≥0, f≥0, and g+f>0, the infinite dynamical behavior of system (5) on the sphere is summarized in Figure 14.Figure 14
Phase portraits for system (5) at infinity. (a) f=0,g>0, (b) f>0,g=0, and (c) f>0,g>0.


### 3.2. The Existence of Infinite Heteroclinic Orbits

Now, we study how the orbits in the vicinity of the Poincaré ball reach to the equilibrium point of the sphere alongz-axis. In fact, we only need to study how the orbits in a neighborhood of 0,0,z30<z3≪1 in system (26) tend to the origin on the chart U3.From the corresponding relation between the local chartU3 and the hemispherical surface, we have(54)y1=z1Δz,y2=z2Δz,y3=Δz,y4=z3Δz,where y4>0 and Δz=1+z12+z22+z32−1/2. Let z1=0, z2=0, and 0<z3≪1; then, y1=y2=0, y3⟶1, and y4⟶0 as z3⟶0. This means that the axial lines in the interior of the Poincaré sphere tend to the endpoint of the positive z-axis. Hence, we get the following results about the infinite equilibria.Lemma 9.
The equilibrium points at the endpoints ofz-axis on the sphere at infinity are asymptotically stable (resp. unstable) along z-axis for b<0 (resp. b>0).Proof.
Since the equilibrium point is at the end point of positive (resp. negative)z-axis on the sphere at infinity coincide with the origin of the Poincaré compactification pX for system (5) in the local chart U3 (resp. V3), we deal with the origin I00,0,0 for system (51) first. Obviously, z3-axis is invariant under the flow of (51); notice, for z1=0 and z2=0, (51) is reduced to(55)z˙1=0,z˙2=0,z˙3=bz32.
Hence,I00,0,0 is asymptotically stable for b<0, which means that orbits tend to I0 along z3-axis from z3>0; I0=0,0,0 is unstable for b>0, which means that orbits are far from I0 along z3-axis from z3>0.
As to the equilibrium point at the endpoint of negativez-axis, we should consider(56)z˙1=0,z˙2=0,z˙3=−bz32,where we need to take into account the case z3<0. The proof can be finished then.
Based on the conclusions obtained above, we may achieve the following result about infinite heteroclinic orbits.Theorem 9.
Ifb≠0, then there exist two infinite heteroclinic orbits denoted by Γi (i=1,2) for system (5): Γ1 connects the origin E0 and the endpoint of the positive z-axis along the positive part of z-axis; Γ2 connects the origin E0 and the endpoint of the negative z-axis along the negative part of z-axis. Furthermore, the origin is unstable for b<0 and stable for b>0, one may see Figure 15.Figure 15
Infinite heteroclinic orbits for system (5) when b<0 and b>0.Proof.
Whenb≠0, c,d∈ℝ, we have presented the dynamical properties of the origin E0 in Table 2. The eigenvalues of Jacobian matrix for system (5) at E0 are(57)λ1,2=c−a±a−c2+4ac−d2,λ3=−b,and the corresponding eigenvectors can be taken as(58)v1,2=−a−c±a−c2+4ac−d−2d,1,0,v3=0,0,1.
Next, we only give the proof of the situationb<0 here; for b>0, we may finish the proof similarly.
Notice that the eigenvalueλ3=−b>0 for b<0, which means that the instability index of E0 is at least 1. It is not difficult to see from system (5) that z-axis is invariant, which directly leads to the instability along z-axis of the equilibrium point E0. Furthermore, we have proved that the equilibrium points at the endpoints of z-axis on the sphere at infinity are asymptotically stable. Consequently, there exist two infinite heteroclinic orbits for system (5), and we may denote them by Γ1 and Γ2, respectively. Moreover, Γ1 connects the origin and the endpoint of the positive z-axis along positive part of the z-axis, while Γ2 connects the origin and the endpoint of the negative z-axis along negative part of the z-axis. These are illustrated in Figure 15.


## 3.1. The Infinite Equilibrium Points of System (5)

To study the dynamics at infinity, we should investigate the Poincaré compactification for system (5) in the local charts Ui and Vii=1,2,3, respectively.


### 3.1.1. In the Local ChartsU1 and V1

Based on [32, 33], the Poincaré compactification pX for system (5) in the chart U1 may be displayed as(41)z˙1=−z2−dz3+a+cz1z3−az12z3,z˙2=f+gz1+a−bz2z3−az1z2z3,z˙3=az32−az1z32.As we know, to study the points of the sphereS2 at infinity, one needs to set z3=0; it then follows that(42)z˙1=−z2,z˙2=f+gz1.One may get that system (42) has no equilibrium point for g=0, while, for g>0, system (42) has a unique equilibrium point −f/g,0. That is to say, system (41) has no equilibrium point for g=0 and has a single equilibrium point P1−f/g,0,0 for g>0. Moreover, a first integral,(43)Hz1,z2=z1+fg2+z22g,has been found in system (42), which shows the equilibrium point P1 of system (41) is a center. And, the phase portrait for (43) has been drawn in Figure 10, which helps us to get the corresponding phase portrait for our system (5) on the sphere at infinity.Figure 10
The phase portrait ofHz1,z2=z1+f/g2+z22/g when f,g=1,1.In order to find out the dynamics of the local property ofP1, we can analyze the flow on the center manifold related to its eigenvalue 0 and achieve the statement listed as follows.Lemma 8.
Wheng>0, the equilibrium point P1 of system (40) has a 1D center manifold Wlocc, and it is unstable along Wlocc.Proof.
Since the equilibrium pointP1 has eigenvalues λ1,2=±gi and λ3=0, which implies P1 has a 1D Wlocc, set k1=z1+f/g, k2=z2, and k3=z3, which translates the singular point P1 into the origin O0,0,0. Then, system (41) is converted into(44)k˙1=−k2−dk3+a+ck1−f/gk3−ak1−f/g2k3,k˙2=gk1+a−bk2k3−ak1−f/gk2k3,k˙3=ak32−ak1−f/gk32.
Thus, the stability ofP1 restricted on the one-dimensional center manifold can be established according to that of the origin on a center manifold. Assume that the graph of a function h:ℝ⟶ℝ2 is presented with k1,k2=hk3=h1k3,h2k3, fulfilling(45)h0=0,0,Dh0=0,−m.
Expandh at k3=0 with Taylor series, which gives(46)h1k3=∑i=2∞aik3i,h2k3=−mk3+∑i=2∞aik3i.
Substituting them into (44) and comparing the coefficients of k3 on both sides, we obtain(47)h1k3=−mbgk32+Ok34,h2k3=a−cmbgk33+Ok34.
Then, we derivek˙3=a1+f/gk32+Ok34, which implies that when z3>0, P1 is local unstable along its center manifold.
Notice that multiplying -1 in the compactified vector fieldpX in U1 can give the compactified vector field pX in V1, so the flow in V1 is the same as the flow in U1 if we reverse the time. We have to point out that z3<0 should be considered in the neighborhood of infinity in V1, see Figure 11.Figure 11
(a) Phase portrait for system (40) when a,b,c,d,f,g=1,1,1,1,1,1. (b) Phase portrait of system (42) with f,g=1,1.
(a)(b)


### 3.1.2. In the Local ChartsU2 and V2

The Poincaré compactificationpX for system (5) in U2 has the following form:(48)z˙1=az3−a+cz1z3+dz12z3+z12z22,z˙2=−b+cz2z3+dz1z2z3+gz1+fz12+z1z22,z˙3=−cz32+z1z2z3+dz1z32.By settingz3=0, system (48) becomes(49)z˙1=z12z2,z˙2=gz1+fz12+z1z22.By some simple computations, one knows that, forg=0 and f>0 or g>0 and f=0, system (49) has a line of equilibrium points 0,z2, while, for g>0 and f>0, system (48) has an equilibrium point −g/f,0 except the line of equilibrium points 0,z2, which means system (48) has a line of equilibria Q10,z2,0 for g=0 and f>0 or g>0 and f=0, while, for g>0 and f>0, system (48) has a line of the equilibrium points Q10,z2,0 and an isolated equilibrium point Q2−g/f,0,0.Moreover, system (49) has a first integral,(50)Hz1,z2=z22+2fz1+gz12,which helps us to get the phase portrait of system (51), one may see Figure 12.Figure 12
The phase portrait for system (49) when f,g=0,1.Similar to the situation ofU1 and V1, we can get the flow of system (5) in V2.


### 3.1.3. In the Local ChartsU3 and V3

Similarly, one may has the Poincaré compactificationpX for system (5) in the local chart U3 listed as(51)z˙1=b−az1z3+az2z3−fz13−gz12z2,z˙2=−dz1z3+b+cz2z3−z1−fz12z2−gz1z22,z˙3=bz32−gz1z2z3−fz12z3,which turns into the following form with z3=0:(52)z˙1=−fz13−gz12z2,z˙2=−z1−fz12z2−gz1z22.It is not difficult to see that there is a first integral in system (52):(53)Hz1,z2=gz22+2fz1z2z12,and then, the phase portrait of system (52) will be achieved, which is shown in Figure 13.Figure 13
The phase portrait for system (27) when f,g=1,1.With similar arguments above, we know that system (51) has a line of the equilibrium points R10,z2,0. Moreover, the dynamics in the local charts V3 may be easily obtained.Summarizing the analysis provided above, one may achieve the following result about the infinite dynamics of system (5) on the sphere.Theorem 8.
Supposeg≥0, f≥0, and g+f>0, the infinite dynamical behavior of system (5) on the sphere is summarized in Figure 14.Figure 14
Phase portraits for system (5) at infinity. (a) f=0,g>0, (b) f>0,g=0, and (c) f>0,g>0.


## 3.1.1. In the Local ChartsU1 and V1

Based on [32, 33], the Poincaré compactification pX for system (5) in the chart U1 may be displayed as(41)z˙1=−z2−dz3+a+cz1z3−az12z3,z˙2=f+gz1+a−bz2z3−az1z2z3,z˙3=az32−az1z32.As we know, to study the points of the sphereS2 at infinity, one needs to set z3=0; it then follows that(42)z˙1=−z2,z˙2=f+gz1.One may get that system (42) has no equilibrium point for g=0, while, for g>0, system (42) has a unique equilibrium point −f/g,0. That is to say, system (41) has no equilibrium point for g=0 and has a single equilibrium point P1−f/g,0,0 for g>0. Moreover, a first integral,(43)Hz1,z2=z1+fg2+z22g,has been found in system (42), which shows the equilibrium point P1 of system (41) is a center. And, the phase portrait for (43) has been drawn in Figure 10, which helps us to get the corresponding phase portrait for our system (5) on the sphere at infinity.Figure 10
The phase portrait ofHz1,z2=z1+f/g2+z22/g when f,g=1,1.In order to find out the dynamics of the local property ofP1, we can analyze the flow on the center manifold related to its eigenvalue 0 and achieve the statement listed as follows.Lemma 8.
Wheng>0, the equilibrium point P1 of system (40) has a 1D center manifold Wlocc, and it is unstable along Wlocc.Proof.
Since the equilibrium pointP1 has eigenvalues λ1,2=±gi and λ3=0, which implies P1 has a 1D Wlocc, set k1=z1+f/g, k2=z2, and k3=z3, which translates the singular point P1 into the origin O0,0,0. Then, system (41) is converted into(44)k˙1=−k2−dk3+a+ck1−f/gk3−ak1−f/g2k3,k˙2=gk1+a−bk2k3−ak1−f/gk2k3,k˙3=ak32−ak1−f/gk32.
Thus, the stability ofP1 restricted on the one-dimensional center manifold can be established according to that of the origin on a center manifold. Assume that the graph of a function h:ℝ⟶ℝ2 is presented with k1,k2=hk3=h1k3,h2k3, fulfilling(45)h0=0,0,Dh0=0,−m.
Expandh at k3=0 with Taylor series, which gives(46)h1k3=∑i=2∞aik3i,h2k3=−mk3+∑i=2∞aik3i.
Substituting them into (44) and comparing the coefficients of k3 on both sides, we obtain(47)h1k3=−mbgk32+Ok34,h2k3=a−cmbgk33+Ok34.
Then, we derivek˙3=a1+f/gk32+Ok34, which implies that when z3>0, P1 is local unstable along its center manifold.
Notice that multiplying -1 in the compactified vector fieldpX in U1 can give the compactified vector field pX in V1, so the flow in V1 is the same as the flow in U1 if we reverse the time. We have to point out that z3<0 should be considered in the neighborhood of infinity in V1, see Figure 11.Figure 11
(a) Phase portrait for system (40) when a,b,c,d,f,g=1,1,1,1,1,1. (b) Phase portrait of system (42) with f,g=1,1.
(a)(b)


## 3.1.2. In the Local ChartsU2 and V2

The Poincaré compactificationpX for system (5) in U2 has the following form:(48)z˙1=az3−a+cz1z3+dz12z3+z12z22,z˙2=−b+cz2z3+dz1z2z3+gz1+fz12+z1z22,z˙3=−cz32+z1z2z3+dz1z32.By settingz3=0, system (48) becomes(49)z˙1=z12z2,z˙2=gz1+fz12+z1z22.By some simple computations, one knows that, forg=0 and f>0 or g>0 and f=0, system (49) has a line of equilibrium points 0,z2, while, for g>0 and f>0, system (48) has an equilibrium point −g/f,0 except the line of equilibrium points 0,z2, which means system (48) has a line of equilibria Q10,z2,0 for g=0 and f>0 or g>0 and f=0, while, for g>0 and f>0, system (48) has a line of the equilibrium points Q10,z2,0 and an isolated equilibrium point Q2−g/f,0,0.Moreover, system (49) has a first integral,(50)Hz1,z2=z22+2fz1+gz12,which helps us to get the phase portrait of system (51), one may see Figure 12.Figure 12
The phase portrait for system (49) when f,g=0,1.Similar to the situation ofU1 and V1, we can get the flow of system (5) in V2.


## 3.1.3. In the Local ChartsU3 and V3

Similarly, one may has the Poincaré compactificationpX for system (5) in the local chart U3 listed as(51)z˙1=b−az1z3+az2z3−fz13−gz12z2,z˙2=−dz1z3+b+cz2z3−z1−fz12z2−gz1z22,z˙3=bz32−gz1z2z3−fz12z3,which turns into the following form with z3=0:(52)z˙1=−fz13−gz12z2,z˙2=−z1−fz12z2−gz1z22.It is not difficult to see that there is a first integral in system (52):(53)Hz1,z2=gz22+2fz1z2z12,and then, the phase portrait of system (52) will be achieved, which is shown in Figure 13.Figure 13
The phase portrait for system (27) when f,g=1,1.With similar arguments above, we know that system (51) has a line of the equilibrium points R10,z2,0. Moreover, the dynamics in the local charts V3 may be easily obtained.Summarizing the analysis provided above, one may achieve the following result about the infinite dynamics of system (5) on the sphere.Theorem 8.
Supposeg≥0, f≥0, and g+f>0, the infinite dynamical behavior of system (5) on the sphere is summarized in Figure 14.Figure 14
Phase portraits for system (5) at infinity. (a) f=0,g>0, (b) f>0,g=0, and (c) f>0,g>0.


## 3.2. The Existence of Infinite Heteroclinic Orbits

Now, we study how the orbits in the vicinity of the Poincaré ball reach to the equilibrium point of the sphere alongz-axis. In fact, we only need to study how the orbits in a neighborhood of 0,0,z30<z3≪1 in system (26) tend to the origin on the chart U3.From the corresponding relation between the local chartU3 and the hemispherical surface, we have(54)y1=z1Δz,y2=z2Δz,y3=Δz,y4=z3Δz,where y4>0 and Δz=1+z12+z22+z32−1/2. Let z1=0, z2=0, and 0<z3≪1; then, y1=y2=0, y3⟶1, and y4⟶0 as z3⟶0. This means that the axial lines in the interior of the Poincaré sphere tend to the endpoint of the positive z-axis. Hence, we get the following results about the infinite equilibria.Lemma 9.
The equilibrium points at the endpoints ofz-axis on the sphere at infinity are asymptotically stable (resp. unstable) along z-axis for b<0 (resp. b>0).Proof.
Since the equilibrium point is at the end point of positive (resp. negative)z-axis on the sphere at infinity coincide with the origin of the Poincaré compactification pX for system (5) in the local chart U3 (resp. V3), we deal with the origin I00,0,0 for system (51) first. Obviously, z3-axis is invariant under the flow of (51); notice, for z1=0 and z2=0, (51) is reduced to(55)z˙1=0,z˙2=0,z˙3=bz32.
Hence,I00,0,0 is asymptotically stable for b<0, which means that orbits tend to I0 along z3-axis from z3>0; I0=0,0,0 is unstable for b>0, which means that orbits are far from I0 along z3-axis from z3>0.
As to the equilibrium point at the endpoint of negativez-axis, we should consider(56)z˙1=0,z˙2=0,z˙3=−bz32,where we need to take into account the case z3<0. The proof can be finished then.
Based on the conclusions obtained above, we may achieve the following result about infinite heteroclinic orbits.Theorem 9.
Ifb≠0, then there exist two infinite heteroclinic orbits denoted by Γi (i=1,2) for system (5): Γ1 connects the origin E0 and the endpoint of the positive z-axis along the positive part of z-axis; Γ2 connects the origin E0 and the endpoint of the negative z-axis along the negative part of z-axis. Furthermore, the origin is unstable for b<0 and stable for b>0, one may see Figure 15.Figure 15
Infinite heteroclinic orbits for system (5) when b<0 and b>0.Proof.
Whenb≠0, c,d∈ℝ, we have presented the dynamical properties of the origin E0 in Table 2. The eigenvalues of Jacobian matrix for system (5) at E0 are(57)λ1,2=c−a±a−c2+4ac−d2,λ3=−b,and the corresponding eigenvectors can be taken as(58)v1,2=−a−c±a−c2+4ac−d−2d,1,0,v3=0,0,1.
Next, we only give the proof of the situationb<0 here; for b>0, we may finish the proof similarly.
Notice that the eigenvalueλ3=−b>0 for b<0, which means that the instability index of E0 is at least 1. It is not difficult to see from system (5) that z-axis is invariant, which directly leads to the instability along z-axis of the equilibrium point E0. Furthermore, we have proved that the equilibrium points at the endpoints of z-axis on the sphere at infinity are asymptotically stable. Consequently, there exist two infinite heteroclinic orbits for system (5), and we may denote them by Γ1 and Γ2, respectively. Moreover, Γ1 connects the origin and the endpoint of the positive z-axis along positive part of the z-axis, while Γ2 connects the origin and the endpoint of the negative z-axis along negative part of the z-axis. These are illustrated in Figure 15.


## 4. Conclusion and the Future Work

The paper proposed a new three-dimensional Lorenz-like chaotic system with eight terms; some interesting and important dynamics of this system were dug out. By combining theoretical analysis and numerical simulations, we rigorously investigated the dynamics at finity and at infinity of system (5). More exactly, the classification of the equilibrium points, the local properties of the nonhyperbolic equilibrium points, the Hopf bifurcations, and the expressions of the local manifolds were clearly provided. As to the dynamics at infinity, we analyzed the flow of the proposed system near and at infinity. Particularly, the existence of infinite heteroclinic orbits was studied by combining the dynamic at finity and infinity.Even so, there are still more important research studies to be carried on in the further work, such as the theoretical proof to the existence of the chaos, chaos control, and applications of our studied system.



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