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We have applied a famous engineering method, called model reference control, to control hyperchaos. We have proposed a general description of the hyperchaotic system and its reference system. By using the Lyapunov stability theorem, we have obtained the expression of the controller. Four examples for the both certain case and the uncertain case show that our method is very effective for controlling hyperchaotic systems with both certain parameters and uncertain parameters.

Chaos has received increasing attentions in the last thirty years. Compared with the ordinary chaotic systems, the hyperchaotic systems hold at least two positive Lyapunov exponents and then possess more complicated attractors. Hyperchaotic systems have the characteristics of high capacity, high security, and high efficiency and have been studied in many fields, such as secure communication [

Controlling chaos (or hyperchaos) is very meaningful. The research has been started since the pioneering work of OGY method [

In this paper, we will show that although a hyperchaotic system is complex, its dynamics can still be controlled along an expected trajectory. The controller is generated by an advantage control method, called model reference control (MRC). Nowadays, MRC is widely used in engineering, such as control of robots [

The first example of the hyperchaotic systems was presented by Rössler in 1979 [

Since a hyperchaotic system with certain parameters should have quadratic terms at least, we may formulate the system as follows:

Equation (

In this section, we will also introduce a reference system with only linear terms and constants. Our aim is to control the hyperchaotic system track along with the reference model system that exhibits asymptotic stability as follows

Particularly, if the reference system has a two-order term, we may write it as

In this section, MRC is applied to control a hyperchaotic system with both certain parameters and uncertain parameters. Our objective is to obtain the exact control law

Let

Assume that the error system (

In the whole process of MRC, the output of reference model and that of the controlled system are compared, and the error vector

If we want to control the system (

We will try to design a control vector

From (

Let

The fact that the hyperchaotic system (

For the symmetric property, the scalar equation (

Using the result of Theorem

If we want to control the system (

It is easy to obtain the control law in the following theorem with similar process of Theorem

Let

We can omit the proof of Theorem

Theorems

Let

For the uncertain property of system (

Hence, the controller

If we want to control the system (

Let

We can also omit the proof of Theorem

This section has two parts. The first part is the numerical examples for controlling hyperchaotic systems, where all the parameters are certain. We will use the result in Theorem

The whole numerical results show that MRC is very suitable and efficient for controlling hyperchaotic system with both certain parameters and uncertain parameters.

The four-variable hyperchaotic Rössler system is described by

The reference model with asymptotical stability is as follows:

Comparison of the corresponding variables in the controlled hyperchaotic Rössler system and its reference system: (a)

The four-variable hyperchaotic Lorenz system [

The reference model with asymptotical stability is as follows:

Figure

Comparison of the corresponding variables in the controlled hyperchaotic Lorenz system and its reference system: (a)

The four-variable hyperchaotic Rössler system with uncertain parameters is described by

According to (

The estimation of the uncertain parameters: (a)

The estimation of

The estimation of

The estimation of

The estimation of

The estimation system of uncertain parameters is

The reference model is the same as Example I,

Figure

Comparison of the corresponding variables in the controlled hyperchaotic R

Comparing with Figures

The four-variable hyperchaotic Lorenz system with uncertain parameters is described by

The estimation of the uncertain parameters: (a)

The estimation of

The estimation of

The estimation of

The estimation of

The estimation of

The estimation of

The estimation system of uncertain parameters is

The reference model is the same as Example II with asymptotical stability is as follows:

Figure

Comparison of the corresponding variables in the controlled hyperchaotic Lorenz system and its reference system: (a)

The performance in Figures

In this paper, we have used an MRC technique to control the hyperchaotic system to track with an expected trajectory. The expression of the controller has been given. Four numerical examples show that the MRC method is very effective for controlling both the hyperchaotic system with all certain parameters and the systems with uncertain parameters. By comparing the results in the corresponding figures, we find that our method does not only fit for controlling hyperchaotic systems with certain parameters, but also is a robust for the systems with uncertain parameters.

This paper is supported by China Scholarship Council. P. F. Zhao expresses sincere thanks to Professor Y. Li for his guidance, Professor D. Liu and Professor B. Yang for their instructions. Also, many thanks to the NSFC (grant number 11071026, 11001100, 11171131, 61133011, 61170092, 60973088, 61202308 and 11026043) and the Basic Research Program of Jilin University (450060481098). This work is supported in part by the National Basic Research Program of China (973) under Grant 2009CB219301, the National Public Benefit Scientific Research Foundation of China under Grant 201011078, the China Scholarship Council and the National Innovation Research Project for Exploration and Development of Oil Shale under Grant OSP-02 and OSR-02. The authors gratefully acknowledge the anonymous reviewers for their hard work and good patience.