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.

Since Lorenz found the first chaotic attractor in the 3D autonomous chaotic system in 1963 [

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 [

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.

In [

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

Here, let

By simple computation, it is easy to obtain that system (

Thus the corresponding characteristic equation can be obtained as

According to the Routh-Hurwitz criterion, the real parts of the roots for (

For the only one equilibrium point

when

when

When

Hyperchaotic attractor of system (

Hyperchaotic attractor in

The Lyapunov exponent spectrum.

Bifurcation diagram of system (

The new hyperchaotic system (

The divergence of system (

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 (

In order to investigate the complex structure and forming mechanism of the new hyperchaotic attractor, its controlled system is proposed and expressed as

It is found that the new hyperchaotic attractor of system (

Left half attractor

Right half attractor

Next, the forming mechanism of the new hyperchaotic attractor will be revealed by changing the value of the constant controller

When

When

When

When

When

When

State trajectory of

Periodic trajectories in

Period-doubling bifurcation in

Hyperchaotic attractor in

Complete hyperchaotic attractor in

The bifurcation diagram of state variable

Bifurcation diagram of system (

In Section

The equilibrium point

Obviously, (

Then, according to (

When

The direction, stability, and period of bifurcating periodic solutions for system (

Define

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

System (

if

if

if

An example of system (

Waveform for system (

Phase portrait of (

Waveform of system (

Phase diagram for (

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.

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).