Journal of Mathematics

Volume 2013 (2013), Article ID 935384, 9 pages

http://dx.doi.org/10.1155/2013/935384

Research Article

Dynamics of a New Hyperchaotic System with Only One Equilibrium Point

Xiang Li and Ranchao Wu

School of Mathematics, Anhui University, Hefei 230039, China

Received 24 April 2013; Revised 27 June 2013; Accepted 2 July 2013

Academic Editor: NanJing Huang

Copyright © 2013 Xiang Li and Ranchao Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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

which is chaotic when , , and .

Based on the Lorenz system, a new four-dimensional system is expressed as

where , and 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 and the Jacobian matrix at the equilibrium point 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 of system (2),

(1)when , the equilibrium is asymptotically stable;(2)when , the equilibrium is unstable.

When , , , , , using MATLAB software, it is easy to show that the eigenvalues are , , , and . 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 with parameter is shown in Figure 4.

Figure 1: Hyperchaotic attractor of system (2).

Figure 2: Hyperchaotic attractor in plane.

Figure 3: The Lyapunov exponent spectrum.

Figure 4: Bifurcation diagram of system (2) versus .

3. Analysis of Basic Dynamic Behaviors

3.1. Symmetry

The new hyperchaotic system (2) is invariant under the coordinate transformation . That is, under reflection about the -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

When , , , so system (2) is a dissipative system and converges with an index rate of . Volume element shrinks to at the time . When , volume element shrinks to . Therefore, all trajectories of the system will be confined to a congregation, whose volume is . 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, , , , , and when , , , , and . The Lyapunov dimension of chaotic attractor of this new hyperchaotic system is fractional, which is described as

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

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 , while one can get the right half attractor when , 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 .

Figure 6: Right half attractor .

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 , , , , and , and its initial values are selected as .

(1)When , the hyperchaotic attractor evolves into the convergent trajectories, which are shown in Figure 7.(2)When , the hyperchaotic attractor evolves into the periodic trajectories, which are shown in Figure 8.(3)When , the hyperchaotic attractor evolves into the period-doubling bifurcations, which are shown in Figure 9.(4)When , the hyperchaotic attractor evolves into left half attractor (right half attractor), which is shown in Figures 5 and 6, respectively.(5)When , the hyperchaotic attractor evolves into partial one but is still bounded in time, which is shown in Figure 10.(6)When , the complete hyperchaotic attractor is obtained and shown in Figure 11.

Figure 7: State trajectory of .

Figure 8: Periodic trajectories in plane .

Figure 9: Period-doubling bifurcation in plane .

Figure 10: Hyperchaotic attractor in plane .

Figure 11: Complete hyperchaotic attractor in plane .

The bifurcation diagram of state variable 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.

Figure 12: Bifurcation diagram of system (2) versus .

5. 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.

5.1. Existence of Hopf Bifurcation

The equilibrium point is asymptotically stable when inequality holds. The critical value for the stability can be derived to be . Let . Equation (5) can be rewritten into

Obviously, (11) has four roots as follows, with a pair being purely imaginary conjugate:

Then, according to (5), one has

Therefore,

Obviously, , so the Hopf bifurcation theorem [18] holds. Hence, when , system (2) undergoes a Hopf bifurcation at the equilibrium point . The above analysis is summarized as follows.

Theorem 2 (existence of the Hopf bifurcation). When passes through the critical value , 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 associated with , , are, respectively,

Define

Then system (2) can be written into

where

in which

In the following, we will follow the procedures proposed by Hassard et al. [18] to figure out the necessary quantities:

where

Through some tedious calculations, one can get

Note that in (20) and (22) it is not necessary to calculate , if one wants to obtain expression of . Its expression is calculated as follows:

in which and are defined to be the same as those in (19).

Based on the previous analysis, one could calculate the following quantities:

where , , and , , 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 as passes through , with the properties below:

(a)if (<0), the Hopf bifurcation is supercritical (subcritical) and bifurcating periodic solutions exist for (<c_0);(b)if (>0), the bifurcating periodic solutions are orbitally stable (unstable);(c)if (<0), the period of bifurcating periodic solutions increases (decreases).

5.3. Numerical Simulations

An example of system (2) is given with , , , and . According to Theorem 2, one has . And it follows from the results in Section 5.2 and some tedious calculations that

In the light of Theorem 3, since , the Hopf bifurcation is supercritical, which means that the equilibrium of system (2) is stable when , as shown in Figures 13 and 14. The equilibrium losses its stability and a Hopf bifurcation occurs when increases past . That is, a family of periodic solutions bifurcate from the equilibrium point and each individual periodic solution is stable for , as shown in Figures 15 and 16. Since , the period of bifurcating periodic solutions decreases with increasing.

Figure 13: Waveform for system (2) with .

Figure 14: Phase portrait of (2) with in () space.

Figure 15: Waveform of system (2) with .

Figure 16: Phase diagram for (2) with in () 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.

Acknowledgments

This work is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20093401120001), the Natural Science Foundation of Anhui Province (no. 11040606 M12), the Natural Science Foundation of Anhui Education Bureau (no. KJ2010A035), and the 211 Project of Anhui University (no. KJJQ1102).

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