# Dynamics of a New Hyperchaotic System with Only One Equilibrium Point

Authors: Xiang Li; Ranchao Wu
Journal: Journal of Mathematics
 (2013)

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## Abstract

A new 4D hyperchaotic system is constructed based on the Lorenz system. The compound structure and forming mechanism of the new hyperchaotic attractor are studied via a controlled system with constant controllers. Furthermore, it is found that the Hopf bifurcation occurs in this
hyperchaotic system when the bifurcation parameter exceeds a critical value. The direction of the Hopf bifurcation as well as the stability of bifurcating periodic solutions is presented in detail by virtue of the normal form theory. Numerical simulations are given to illustrate and verify the results.


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## Body

## 1. Introduction

Since Lorenz found the first chaotic attractor in the 3D autonomous chaotic system in 1963 [1], people have realized that chaos is a ubiquitous and extremely complex nonlinear phenomenon in nature. In the past few years, motivated by many unknown interesting properties and some potential practical applications, great efforts have been made in constructing chaotic and hyperchaotic systems. Particularly, Chen and Ueta found a new chaotic system, called the Chen system [2] in 1999, based on Lorenz system. Afterwards, Lü and Chen furthermore found a chaotic system [3], which represents the transition between the Lorenz system and the Chen system. Some other new chaotic systems, including the Liu system [4, 5] and T system [6], have also been constructed and investigated in recent years.Recently, applications of hyperchaos have become a central topic in research. Some interesting hyperchaotic systems were presented in the past two decades, and their dynamics have been investigated extensively owing to their useful potential applications in engineering. Historically, hyperchaos was firstly reported by Rössler in 1979 [7], which was the noted 4D hyperchaotic Rössler system. A hyperchaotic system is usually defined as a chaotic system with more than one positive Lyapunov exponent. It means that hyperchaotic systems have more complex dynamical behaviors than chaotic systems [8–14]. These complex dynamics could be explored via bifurcation analysis of systems with varying parameters. Nowadays, bifurcation is one of most active research topics in the field of nonlinear science [15–17].Now in this paper, based on the Lorenz system, a new four-dimensional hyperchaotic system with only one equilibrium point is constructed. Some basic dynamical properties, such as the Lyapunov exponents, bifurcation diagram, fractal dimensions, and hyperchaotic behaviors of this new system are investigated. Furthermore, the compound structure and forming mechanism of the new hyperchaotic attractor are studied by a controlled system with constant controllers. It is found that the two single scroll attractors, which form the complete compound hyperchaotic attractor, merely originate from some simple limit circles. As is well known, the Hopf bifurcations can give rise to limit circles. Therefore, the Hopf bifurcation analysis is carried out to investigate its complex dynamical behaviors. See that a Hopf bifurcation occurs in this system when the bifurcation parameter exceeds a critical value. The direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are also presented by applying the normal form theory.


## 2. A New 4D Hyperchaotic System

In [1], the Lorenz system is given by
(1)x˙=a(y-x),y˙=cx-xz-y,z˙=xy-bz,
which is chaotic when a=10, b=8/3, and c=28.Based on the Lorenz system, a new four-dimensional system is expressed as(2)x˙1=a(x2-x1),x˙2=cx1-x1x3-x2+ex4,x˙3=x14+x22-bx3,x˙4=-dx2,
where x1,x2,x3, and x4 are state variables and a,b,c,d, and e are parameters.Here, let(3)a>0,b>0,de>0.By simple computation, it is easy to obtain that system (2) has only one equilibrium point O(0,0,0,0) and the Jacobian matrix J at the equilibrium point O(0,0,0,0) is
(4)J=(-aa00c-10e00-b00-d00).Thus the corresponding characteristic equation can be obtained as(5)f(λ)=(λ+b)×[λ3+(a+1)λ2+(a+de-ac)λ+ade]=0.According to the Routh-Hurwitz criterion, the real parts of the roots for (5) are all negative if and only if parameters satisfy the condition
(6)c<1+de(a2+a).
Thus, one can get the following.Theorem 1.
For the only one equilibrium pointO(0,0,0,0) of system (2),(1)
whenc<1+de/(a2+a), the equilibrium O is asymptotically stable;(2)
whenc≥1+de/(a2+a), the equilibrium O is unstable.Whena=10, b=8/3, c=28, d=2, e=12, using MATLAB software, it is easy to show that the eigenvalues are λ1=-22.4395, λ2=10.4123, λ3=1.0272, and λ4=-2.6667. Hence, the equilibrium O is an unstable saddle, and the four Lyapunov exponents are, respectively, to be
(7)λ1=0.8331,λ2=0.1703,λ3=0,λ4=-1.4456.
Obviously, there are two positive Lyapunov exponents. Therefore, the new 4D system (2) is a hyperchaotic system, and the hyperchaotic attractor is shown in Figures 1 and 2. The Lyapunov exponent spectrum is shown in Figure 3, and the bifurcation diagram of state variable x2 with parameter c is shown in Figure 4.Figure 1
Hyperchaotic attractor of system (2).Figure 2
Hyperchaotic attractor inx1-x2 plane.Figure 3
The Lyapunov exponent spectrum.Figure 4
Bifurcation diagram of system (2) versus c.


## 3. Analysis of Basic Dynamic Behaviors

### 3.1. Symmetry

The new hyperchaotic system (2) is invariant under the coordinate transformation (x1,x2,x3,x4)→(-x1,-x2,x3,-x4). That is, under reflection about the z-axis, the symmetry persists for all values of the system parameters.


### 3.2. Dissipation and the Existence of an Attractor

The divergence of system (2) is defined by
(8)∇V=∂x˙1∂x1+∂x˙2∂x2+∂x˙3∂x3+∂x˙4∂x4=-a-1-b.
When a=10, b=8/3, ∇V=-41/3<0, so system (2) is a dissipative system and converges with an index rate of e-(41/3)t. Volume element V0 shrinks to V0e-(41/3)t at the time t. When t→∞, volume element V0 shrinks to 0. Therefore, all trajectories of the system will be confined to a congregation, whose volume is 0. And the gradual movement behaviors are fixed into an attractor.


### 3.3. The Lyapunov Dimension

As we know, the Lyapunov exponents measure the exponential rates of divergence or convergence of nearby trajectories in phase space. The four Lyapunov exponents of system (2) are, respectively, λ1=0.8331, λ2=0.1703, λ3=0, λ4=-1.4456, and when a=10, b=8/3, c=28, d=2, and e=12. The Lyapunov dimension of chaotic attractor of this new hyperchaotic system is fractional, which is described as
(9)DL=j+1|λj+1|∑i=1j‍λi=3+1|-1.4456|(0.8331+0.1703+0)=3.6941.


## 3.1. Symmetry

The new hyperchaotic system (2) is invariant under the coordinate transformation (x1,x2,x3,x4)→(-x1,-x2,x3,-x4). That is, under reflection about the z-axis, the symmetry persists for all values of the system parameters.


## 3.2. Dissipation and the Existence of an Attractor

The divergence of system (2) is defined by
(8)∇V=∂x˙1∂x1+∂x˙2∂x2+∂x˙3∂x3+∂x˙4∂x4=-a-1-b.
When a=10, b=8/3, ∇V=-41/3<0, so system (2) is a dissipative system and converges with an index rate of e-(41/3)t. Volume element V0 shrinks to V0e-(41/3)t at the time t. When t→∞, volume element V0 shrinks to 0. Therefore, all trajectories of the system will be confined to a congregation, whose volume is 0. And the gradual movement behaviors are fixed into an attractor.


## 3.3. The Lyapunov Dimension

As we know, the Lyapunov exponents measure the exponential rates of divergence or convergence of nearby trajectories in phase space. The four Lyapunov exponents of system (2) are, respectively, λ1=0.8331, λ2=0.1703, λ3=0, λ4=-1.4456, and when a=10, b=8/3, c=28, d=2, and e=12. The Lyapunov dimension of chaotic attractor of this new hyperchaotic system is fractional, which is described as
(9)DL=j+1|λj+1|∑i=1j‍λi=3+1|-1.4456|(0.8331+0.1703+0)=3.6941.


## 4. Complexity and Forming Mechanism of the Attractor

In order to investigate the complex structure and forming mechanism of the new hyperchaotic attractor, its controlled system is proposed and expressed as(10)x˙1=a(x2-x1),x˙2=cx1-x1x3-x2+ex4,x˙3=x14+x22-bx3,x˙4=-dx2+u,
where u is a constant controller, which can effectively control the dynamical behavior of the new hyperchaotic system.It is found that the new hyperchaotic attractor of system (2) is evolved into the left half attractor when u=-2.9, while one can get the right half attractor when u=2.9, which is the mirror operation of the left half attractor. It means that the new hyperchaotic attractor of system (2) has a compound structure. That is, it can be obtained by merging together two single scroll attractors (left half attractor and right half attractor) which are shown in Figures 5 and 6, respectively, after performing one mirror operation.Figure 5
Left half attractor(u=-2.9).Figure 6
Right half attractor(u=2.9).Next, the forming mechanism of the new hyperchaotic attractor will be revealed by changing the value of the constant controlleru within a certain range. Here, the parameters of the controlled system (10) are chosen as a=10, b=8/3, c=28, d=2, and e=12, and its initial values are selected as (0,1,0,0).(1)
When|u|≥9.4, the hyperchaotic attractor evolves into the convergent trajectories, which are shown in Figure 7.(2)
When3.14<|u|≤9.3, the hyperchaotic attractor evolves into the periodic trajectories, which are shown in Figure 8.(3)
When2.9<|u|≤3.14, the hyperchaotic attractor evolves into the period-doubling bifurcations, which are shown in Figure 9.(4)
When2.8<|u|≤2.9, the hyperchaotic attractor evolves into left half attractor (right half attractor), which is shown in Figures 5 and 6, respectively.(5)
When0.5<|u|≤2.8, the hyperchaotic attractor evolves into partial one but is still bounded in time, which is shown in Figure 10.(6)
When|u|≤0.5, the complete hyperchaotic attractor is obtained and shown in Figure 11.Figure 7
State trajectory ofx2(u=10).Figure 8
Periodic trajectories inx1-x2 plane (u=3.5).Figure 9
Period-doubling bifurcation inx1-x2 plane (u=3.1).Figure 10
Hyperchaotic attractor inx1-x2 plane (u=1.5).Figure 11
Complete hyperchaotic attractor inx1-x2 plane (u=0.2).The bifurcation diagram of state variablex2 versus constant controller u is shown in Figure 12. It is easy to see that the hyperchaotic attractor disappears when |u| is large enough; when |u| is small enough, a complete hyperchaotic attractor appears. Meanwhile, it has been found that its compound structure can be obtained by merging together two single scroll attractors after performing one mirror operation. However, the two single scroll attractors merely originate from some simple limit circles as shown in Figure 8 and are obtained after the period-doubling bifurcations.Figure 12
Bifurcation diagram of system (2) versus u.


## 5. Hopf Bifurcation in the New Hyperchaotic System

In Section4, the compound structure and forming mechanism of the new hyperchaotic attractor are studied via detailed numerical simulations as well as theoretical analysis. It is shown that some simple limit circles are very important for evolution of single scroll chaotic attractors. Since the Hopf bifurcations can give rise to limit circles, then the Hopf bifurcation in the new hyperchaotic system (2) will be further discussed in this section.


### 5.1. Existence of Hopf Bifurcation

The equilibrium pointO(0,0,0,0) is asymptotically stable when inequality c<1+de/(a2+a) holds. The critical value for the stability can be derived to be c0=1+de/(a2+a). Let c=c0. Equation (5) can be rewritten into
(11)(λ+b)(λ+a+1)(λ2+adea+1)=0.Obviously, (11) has four roots as follows, with a pair being purely imaginary conjugate:
(12)λ1=iadea+1=iω0,λ2=-iadea+1=-iω0,λ3=-(a+1),λ4=-b.Then, according to (5), one has
(13)λ′(c)=(aλ(λ+bλ))×(4λ3+(a+b+1)λ2+2(a+de-ac+ab+b)λ+ab(1-c)+de(a+b)4λ3+(a+b+1)λ2)-1.
Therefore,
(14)Re(λ′(c0))=(ab2(a+1)3+a2de(a+1)2)×(2[b2(a+1)4+ade(a+1)3==+abde(a+1)2+ab2de(a+1)+a2d2e2])-1,Im(λ′(c0))=([ab2(a+1)2+a2de(a+1)]ω0)×(2[b2(a+1)4+ade(a+1)3=×+abde(a+1)2=×.+ab2de(a+1)+a2d2e2])-1.
Obviously, Re(λ′(c0))≠0, so the Hopf bifurcation theorem [18] holds. Hence, when c=c0=1+de/(a2+a), system (2) undergoes a Hopf bifurcation at the equilibrium point O(0,0,0,0). The above analysis is summarized as follows.Theorem 2 (existence of the Hopf bifurcation).
Whenc passes through the critical value c0=1+de/(a2+a), system (2) undergoes a Hopf bifurcation at the equilibrium point.


### 5.2. Direction and Stability of Bifurcating Periodic Solutions

The direction, stability, and period of bifurcating periodic solutions for system (2) will be investigated in detail by virtue of the normal form theory [18]. The eigenvectors υ1,υ3,υ4 associated with λ1=iω0, λ3=-(a+1), λ4=-b are, respectively,(15)υ1=(-ade+iaade(a+1)a(a2+a+de)iade(a+1)d(a+1)01),υ3=(-a(a+1)da+1d01),υ4=(0010).Define(16)P=(Reυ1,Imυ1,υ3,υ4)=(-ae(a2+a+de)aade(a+1)d(a2+a+de)-a(a+1)d00ade(a+1)d(a+1)a+1d000011010),(x1,x2,x3,x4)T=P(y1,y2,y3,y4)T.
Then system (2) can be written into
(17)y˙1=-ω0y2+F1(y1,y2,y3,y4),y˙2=ω0y1+F2(y1,y2,y3,y4),y˙3=-(a+1)y3+F3(y1,y2,y3,y4),y˙4=-by4+F4(y1,y2,y3,y4),
where
(18)F1(y1,y2,y3,y4)=-ry4(ly1+my2+ny3)+er(y1+y3),F2(y1,y2,y3,y4)=-sy4(ly1+my2+ny3)+er(y1+y3),F3(y1,y2,y3,y4)=-F1(y1,y2,y3,y4),F4(y1,y2,y3,y4)=(ly1+my2+ny3)4+(py2+qy3)2,
in which
(19)l=-aea2+a+de,m=aade(a+1)d(a2+a+de),n=-a(a+1)d,p=ade(a+1)d(a+1),r=-d(a+1)a3+3a2+3a+ade+1,q=a+1d,s=ad(a+1)(a2+2a+de+1)ade(a+1)(a3+3a2+3a+ade+1).
In the following, we will follow the procedures proposed by Hassard et al. [18] to figure out the necessary quantities:
(20)g20=14[∂2F1∂y12-∂2F1∂y22+2∂2F2∂y1∂y2+i(∂2F2∂y12-∂2F2∂y22-2∂2F1∂y1∂y2)],g11=14[∂2F1∂y12+∂2F1∂y22+i(∂2F2∂y12+∂2F2∂y22)],g02=14[∂2F1∂y12-∂2F1∂y22-2∂2F2∂y1∂y2+i(∂2F2∂y12-∂2F2∂y22+2∂2F1∂y1∂y2)],g21=G21+2G1101w111+G1011w201+2G1102w112+G1012w202,
where
(21)G21=18[∂3F1∂y13+∂3F1∂y1∂y22+∂3F2∂y12∂y2+∂3F2∂y23+i(∂3F2∂y13+∂3F2∂y1∂y22-∂3F1∂y12∂y2-∂3F1∂y23)],G1101=12[∂2F1∂y1∂y3+∂2F2∂y2∂y3+i(∂2F2∂y1∂y3-∂2F1∂y2∂y3)],G1102=12[∂2F1∂y1∂y4+∂2F2∂y2∂y4+i(∂2F2∂y1∂y4-∂2F1∂y2∂y4)],G1011=12[∂2F1∂y1∂y3-∂2F2∂y2∂y3+i(∂2F2∂y1∂y3+∂2F1∂y2∂y3)],G1012=12[∂2F1∂y1∂y4-∂2F2∂y2∂y4+i(∂2F2∂y1∂y4+∂2F1∂y2∂y4)],w111=-14λ3(c0)(∂2F3∂y12+∂2F3∂y22),w112=-14λ4(c0)(∂2F4∂y12+∂2F4∂y22),w201=14(2iω0-λ3(c0))(∂2F3∂y12-∂2F3∂y22-2i∂2F3∂y1∂y2),w202=14(2iω0-λ3(c0))(∂2F4∂y12-∂2F4∂y22-2i∂2F4∂y1∂y2),C1(0)=i2ω0[g20g11-2|g11|2-13|g02|2]+12g21.
Through some tedious calculations, one can get
(22)g20=g11=g02=G21=G1101=w201=0,(23)G1102=-12(rl+sm)+i2(rm-sl),w112=-p22λ4,G1012=-12(rl-sm)+i2(rm+sl),w202=-p22(2iω0-λ3).
Note that in (20) and (22) it is not necessary to calculate w111,G1011, if one wants to obtain expression of g21. Its expression is calculated as follows:
(24)g21=(p2(rl+sm)(2λ32+8ω02+2λ4ω0)+λ3λ4p2(rl-sm))×(4λ4(λ32+4ω02))-1+i(p2(rl+sm)(2λ32+8ω02)+i+i+2λ4ω0p2(rl-sm)-λ3λ4p2(sl-rm))×(4λ4(λ32+4ω02))-1,
in which i2=-1 and l,m,n,p,r,s are defined to be the same as those in (19).Based on the previous analysis, one could calculate the following quantities:(25)C1(0)=12g21=(p2(rl+sm)(2λ32+8ω02+2λ4ω0)=+=+λ3λ4p2(rl-sm))×(2λ4(λ32+4ω02))-1+i(p2(rl+sm)(2λ32+8ω02)+i+2λ4ω0p2(rl-sm)-λ3λ4p2(sl-rm))×(2λ4(λ32+4ω02))-1,μ2=-ReC1(0)α′(0)=-(p2(rl+sm)(2λ32+8ω02+2λ4ω0)+λ3λ4p2(rl-sm))×(λ4(λ32+4ω02))-1×([b2(a+1)4+ade(a+1)3+abde(a+1)2+ab2de(a+1)+a2d2e2])×(ab2(a+1)3+a2de(a+1)2)-1,τ2=-(ImC1(0)+μ2ω′(0))ω0=(p2(rl+sm)==×[2ω02λ32+2λ4ω03+8ω04-(a+1)(2λ32+8ω02)])×(2λ4ω0(a+1)(4ω02+λ32))-1-(p2(rl-sm)[λ3λ4ω02-2λ4ω0(a+1)]====+λ3λ4p2(a+1)(sl-rm))×(2λ4ω0(a+1)(4ω02+λ32))-1,β2=2ReC1(0)=(p2(rl+sm)(2λ32+8ω02+2λ4ω0)=gg+λ3λ4p2(rl-sm))×(4λ4(λ32+4ω02))-1,
where α′(0)=Re(λ′(c0)), ω′(0)=Im(λ′(c0)), and λ3, λ4, are those given in Section 5.2. Now one arrives at the following result.Theorem 3.
System (2) exhibits a Hopf bifurcation at the equilibrium O(0,0,0,0) as c passes through c0, with the properties below:(a)
ifμ2>0 (<0), the Hopf bifurcation is supercritical (subcritical) and bifurcating periodic solutions exist for c>c0 (<c_0);(b)
ifβ2<0 (>0), the bifurcating periodic solutions are orbitally stable (unstable);(c)
ifτ2>0 (<0), the period of bifurcating periodic solutions increases (decreases).


### 5.3. Numerical Simulations

An example of system (2) is given with a=10, b=8/3, d=2, and e=12. According to Theorem 2, one has c0=1.2182. And it follows from the results in Section 5.2 and some tedious calculations that
(26)μ2=0.9249,β2=-0.2033,τ2=-0.0429.
In the light of Theorem 3, since μ2>0, the Hopf bifurcation is supercritical, which means that the equilibrium O(0,0,0,0) of system (2) is stable when c<c0, as shown in Figures 13 and 14. The equilibrium losses its stability and a Hopf bifurcation occurs when c increases past c0. That is, a family of periodic solutions bifurcate from the equilibrium point and each individual periodic solution is stable for β2<0, as shown in Figures 15 and 16. Since τ2<0, the period of bifurcating periodic solutions decreases with c increasing.Figure 13
Waveform for system (2) with c=1.Figure 14
Phase portrait of (2) with c=1 in (x1,x2,x4) space.Figure 15
Waveform of system (2) with c=2.5.Figure 16
Phase diagram for (2) with c=2.5 in (x1,x2,x4) space.


## 5.1. Existence of Hopf Bifurcation

The equilibrium pointO(0,0,0,0) is asymptotically stable when inequality c<1+de/(a2+a) holds. The critical value for the stability can be derived to be c0=1+de/(a2+a). Let c=c0. Equation (5) can be rewritten into
(11)(λ+b)(λ+a+1)(λ2+adea+1)=0.Obviously, (11) has four roots as follows, with a pair being purely imaginary conjugate:
(12)λ1=iadea+1=iω0,λ2=-iadea+1=-iω0,λ3=-(a+1),λ4=-b.Then, according to (5), one has
(13)λ′(c)=(aλ(λ+bλ))×(4λ3+(a+b+1)λ2+2(a+de-ac+ab+b)λ+ab(1-c)+de(a+b)4λ3+(a+b+1)λ2)-1.
Therefore,
(14)Re(λ′(c0))=(ab2(a+1)3+a2de(a+1)2)×(2[b2(a+1)4+ade(a+1)3==+abde(a+1)2+ab2de(a+1)+a2d2e2])-1,Im(λ′(c0))=([ab2(a+1)2+a2de(a+1)]ω0)×(2[b2(a+1)4+ade(a+1)3=×+abde(a+1)2=×.+ab2de(a+1)+a2d2e2])-1.
Obviously, Re(λ′(c0))≠0, so the Hopf bifurcation theorem [18] holds. Hence, when c=c0=1+de/(a2+a), system (2) undergoes a Hopf bifurcation at the equilibrium point O(0,0,0,0). The above analysis is summarized as follows.Theorem 2 (existence of the Hopf bifurcation).
Whenc passes through the critical value c0=1+de/(a2+a), system (2) undergoes a Hopf bifurcation at the equilibrium point.


## 5.2. Direction and Stability of Bifurcating Periodic Solutions

The direction, stability, and period of bifurcating periodic solutions for system (2) will be investigated in detail by virtue of the normal form theory [18]. The eigenvectors υ1,υ3,υ4 associated with λ1=iω0, λ3=-(a+1), λ4=-b are, respectively,(15)υ1=(-ade+iaade(a+1)a(a2+a+de)iade(a+1)d(a+1)01),υ3=(-a(a+1)da+1d01),υ4=(0010).Define(16)P=(Reυ1,Imυ1,υ3,υ4)=(-ae(a2+a+de)aade(a+1)d(a2+a+de)-a(a+1)d00ade(a+1)d(a+1)a+1d000011010),(x1,x2,x3,x4)T=P(y1,y2,y3,y4)T.
Then system (2) can be written into
(17)y˙1=-ω0y2+F1(y1,y2,y3,y4),y˙2=ω0y1+F2(y1,y2,y3,y4),y˙3=-(a+1)y3+F3(y1,y2,y3,y4),y˙4=-by4+F4(y1,y2,y3,y4),
where
(18)F1(y1,y2,y3,y4)=-ry4(ly1+my2+ny3)+er(y1+y3),F2(y1,y2,y3,y4)=-sy4(ly1+my2+ny3)+er(y1+y3),F3(y1,y2,y3,y4)=-F1(y1,y2,y3,y4),F4(y1,y2,y3,y4)=(ly1+my2+ny3)4+(py2+qy3)2,
in which
(19)l=-aea2+a+de,m=aade(a+1)d(a2+a+de),n=-a(a+1)d,p=ade(a+1)d(a+1),r=-d(a+1)a3+3a2+3a+ade+1,q=a+1d,s=ad(a+1)(a2+2a+de+1)ade(a+1)(a3+3a2+3a+ade+1).
In the following, we will follow the procedures proposed by Hassard et al. [18] to figure out the necessary quantities:
(20)g20=14[∂2F1∂y12-∂2F1∂y22+2∂2F2∂y1∂y2+i(∂2F2∂y12-∂2F2∂y22-2∂2F1∂y1∂y2)],g11=14[∂2F1∂y12+∂2F1∂y22+i(∂2F2∂y12+∂2F2∂y22)],g02=14[∂2F1∂y12-∂2F1∂y22-2∂2F2∂y1∂y2+i(∂2F2∂y12-∂2F2∂y22+2∂2F1∂y1∂y2)],g21=G21+2G1101w111+G1011w201+2G1102w112+G1012w202,
where
(21)G21=18[∂3F1∂y13+∂3F1∂y1∂y22+∂3F2∂y12∂y2+∂3F2∂y23+i(∂3F2∂y13+∂3F2∂y1∂y22-∂3F1∂y12∂y2-∂3F1∂y23)],G1101=12[∂2F1∂y1∂y3+∂2F2∂y2∂y3+i(∂2F2∂y1∂y3-∂2F1∂y2∂y3)],G1102=12[∂2F1∂y1∂y4+∂2F2∂y2∂y4+i(∂2F2∂y1∂y4-∂2F1∂y2∂y4)],G1011=12[∂2F1∂y1∂y3-∂2F2∂y2∂y3+i(∂2F2∂y1∂y3+∂2F1∂y2∂y3)],G1012=12[∂2F1∂y1∂y4-∂2F2∂y2∂y4+i(∂2F2∂y1∂y4+∂2F1∂y2∂y4)],w111=-14λ3(c0)(∂2F3∂y12+∂2F3∂y22),w112=-14λ4(c0)(∂2F4∂y12+∂2F4∂y22),w201=14(2iω0-λ3(c0))(∂2F3∂y12-∂2F3∂y22-2i∂2F3∂y1∂y2),w202=14(2iω0-λ3(c0))(∂2F4∂y12-∂2F4∂y22-2i∂2F4∂y1∂y2),C1(0)=i2ω0[g20g11-2|g11|2-13|g02|2]+12g21.
Through some tedious calculations, one can get
(22)g20=g11=g02=G21=G1101=w201=0,(23)G1102=-12(rl+sm)+i2(rm-sl),w112=-p22λ4,G1012=-12(rl-sm)+i2(rm+sl),w202=-p22(2iω0-λ3).
Note that in (20) and (22) it is not necessary to calculate w111,G1011, if one wants to obtain expression of g21. Its expression is calculated as follows:
(24)g21=(p2(rl+sm)(2λ32+8ω02+2λ4ω0)+λ3λ4p2(rl-sm))×(4λ4(λ32+4ω02))-1+i(p2(rl+sm)(2λ32+8ω02)+i+i+2λ4ω0p2(rl-sm)-λ3λ4p2(sl-rm))×(4λ4(λ32+4ω02))-1,
in which i2=-1 and l,m,n,p,r,s are defined to be the same as those in (19).Based on the previous analysis, one could calculate the following quantities:(25)C1(0)=12g21=(p2(rl+sm)(2λ32+8ω02+2λ4ω0)=+=+λ3λ4p2(rl-sm))×(2λ4(λ32+4ω02))-1+i(p2(rl+sm)(2λ32+8ω02)+i+2λ4ω0p2(rl-sm)-λ3λ4p2(sl-rm))×(2λ4(λ32+4ω02))-1,μ2=-ReC1(0)α′(0)=-(p2(rl+sm)(2λ32+8ω02+2λ4ω0)+λ3λ4p2(rl-sm))×(λ4(λ32+4ω02))-1×([b2(a+1)4+ade(a+1)3+abde(a+1)2+ab2de(a+1)+a2d2e2])×(ab2(a+1)3+a2de(a+1)2)-1,τ2=-(ImC1(0)+μ2ω′(0))ω0=(p2(rl+sm)==×[2ω02λ32+2λ4ω03+8ω04-(a+1)(2λ32+8ω02)])×(2λ4ω0(a+1)(4ω02+λ32))-1-(p2(rl-sm)[λ3λ4ω02-2λ4ω0(a+1)]====+λ3λ4p2(a+1)(sl-rm))×(2λ4ω0(a+1)(4ω02+λ32))-1,β2=2ReC1(0)=(p2(rl+sm)(2λ32+8ω02+2λ4ω0)=gg+λ3λ4p2(rl-sm))×(4λ4(λ32+4ω02))-1,
where α′(0)=Re(λ′(c0)), ω′(0)=Im(λ′(c0)), and λ3, λ4, are those given in Section 5.2. Now one arrives at the following result.Theorem 3.
System (2) exhibits a Hopf bifurcation at the equilibrium O(0,0,0,0) as c passes through c0, with the properties below:(a)
ifμ2>0 (<0), the Hopf bifurcation is supercritical (subcritical) and bifurcating periodic solutions exist for c>c0 (<c_0);(b)
ifβ2<0 (>0), the bifurcating periodic solutions are orbitally stable (unstable);(c)
ifτ2>0 (<0), the period of bifurcating periodic solutions increases (decreases).


## 5.3. Numerical Simulations

An example of system (2) is given with a=10, b=8/3, d=2, and e=12. According to Theorem 2, one has c0=1.2182. And it follows from the results in Section 5.2 and some tedious calculations that
(26)μ2=0.9249,β2=-0.2033,τ2=-0.0429.
In the light of Theorem 3, since μ2>0, the Hopf bifurcation is supercritical, which means that the equilibrium O(0,0,0,0) of system (2) is stable when c<c0, as shown in Figures 13 and 14. The equilibrium losses its stability and a Hopf bifurcation occurs when c increases past c0. That is, a family of periodic solutions bifurcate from the equilibrium point and each individual periodic solution is stable for β2<0, as shown in Figures 15 and 16. Since τ2<0, the period of bifurcating periodic solutions decreases with c increasing.Figure 13
Waveform for system (2) with c=1.Figure 14
Phase portrait of (2) with c=1 in (x1,x2,x4) space.Figure 15
Waveform of system (2) with c=2.5.Figure 16
Phase diagram for (2) with c=2.5 in (x1,x2,x4) space.


## 6. Conclusions

In this paper, a new 4D hyperchaotic system with only one equilibrium point is presented based on the Lorenz system. Some basic dynamical properties, such as the Lyapunov exponents, bifurcation diagram, fractal dimensions, and hyperchaotic behaviors are investigated. Furthermore, the compound structure and forming mechanism of the new hyperchaotic attractor are revealed via a controlled system with constant controllers. Consequently, it is shown that the new hyperchaotic attractor has a compound structure which can be obtained by merging together two single scroll attractors after performing one mirror operation. Moreover, it is also found that the two single scroll attractors merely originate from some simple limit circles. In addition, the Hopf bifurcation occurs in this system when the bifurcation parameter exceeds a critical value. The direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are analyzed in detail. Finally, some numerical simulations are also carried out to illustrate the results. There are still some interesting dynamical behaviors about this system, which deserve to be further investigated. It is believed that the system will have some useful applications in various chaos-based systems.



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*Source: 935384-2013-07-28.xml*