Dynamics of a New Hyperchaotic System with Only One Equilibrium Point

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.


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][9][10][11][12][13][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][16][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.
Based on the Lorenz system, a new four-dimensional system is expressed as where  1 ,  2 ,  3 , and  4 are state variables and , , , , and  are parameters.
Here, let By simple computation, it is easy to obtain that system (2) has only one equilibrium point (0, 0, 0, 0) and the Jacobian matrix  at the equilibrium point (0, 0, 0, 0) is Thus the corresponding characteristic equation can be obtained as 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 Thus, one can get the following.Theorem 1.For the only one equilibrium point (0, 0, 0, 0) of system (2), (1) when  < 1 + /( 2 + ), the equilibrium  is asymptotically stable; (2) when  ≥ 1+/( 2 +), the equilibrium  is unstable.
When  = 10,  = 8/3,  = 28,  = 2,  = 12, using MATLAB software, it is easy to show that the eigenvalues are  1 = −22.4395, 2 = 10.4123, 3 = 1.0272, and  4 = −2.6667.Hence, the equilibrium  is an unstable saddle, and the four Lyapunov exponents are, respectively, to be 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  2 with parameter  is shown in Figure 4.
When  → ∞, volume element  0 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.
The Lyapunov dimension of chaotic attractor of this new hyperchaotic system is fractional, which is described as

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 where  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  = −2.9, while one can get the right half attractor when  = 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.
Next, the forming mechanism of the new hyperchaotic attractor will be revealed by changing the value of the constant controller  within a certain range.Here, the parameters of the controlled system (10) are chosen as  = 10,  = 8/3,  = 28,  = 2, and  = 12, and its initial values are selected as (0, 1, 0, 0).
The bifurcation diagram of state variable  2 versus constant controller  is shown in Figure 12.It is easy to see that the hyperchaotic attractor disappears when || is large enough; when || 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.

Hopf Bifurcation in the New Hyperchaotic System
In Section 4, 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.

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  1 ,  3 ,  4 associated with  1 =  0 ,  3 = −( + 1),  4 = − are, respectively,  + 1  0 Then system (2) can be written into where ( In the following, we will follow the procedures proposed by Hassard et al. [18] to figure out the necessary quantities: )] , 21 =  21 + 2 where ) , in which  2 = −1 and , , , , ,  are defined to be the same as those in (19).

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 chaosbased systems.