The robust synchronization of hyperchaotic systems with uncertainties and external disturbances is investigated. Based on the Lyapunov stability theory, the appropriate adaptive controllers and parameter update laws are designed to achieve the synchronization of uncertain hyperchaotic systems. The robust synchronization of two hyperchaotic Chen systems is taken as an example to verify the feasibility of the presented schemes. The size of the subcontroller gain’s influences on the convergence speed is discussed. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed synchronization schemes.

Since the method of synchronization between identical chaotic systems with different initial conditions was presented by Pecora and Carroll [

However, most of the reported schemes are mainly concerned with the synchronization of chaotic systems without uncertainties and external disturbances. This behaviour results in that the complexity of chaotic dynamics is limited while, in real world applications, there exist a mass of phenomena of the uncertainties and external disturbances in chaotic systems. In this regard, a number of researchers have paid their attention to the synchronization of chaotic systems with uncertainties and external disturbances [

The chaos-hyperchaos transition occurs when the second Lyapunov exponent becomes positive [

In this paper, depending on the characteristics of external disturbance signals, appropriate adaptive controllers and parameter update laws are proposed for the robust synchronization of hyperchaotic systems with uncertainties and external disturbances. The uncertainties are bounded by a nonlinear state-dependent function, instead of a real constant. In the first adaptive controller, assuming that external disturbances are square integrable signals, the adaptive-based controllers and parameter update laws are designed to remove the effects of uncertainties. In the second proposed adaptive controller, the uncertainties and the external disturbances are not square integrable signals. The uncertainties and external disturbances are combined into an uncertain time-varying function with unknown bound. The robustness properties with respect to uncertainties and external disturbances are provided by the proposed controllers.

The organization of this paper is as follows. In Section

Throughout the paper, for a vector

Consider the nonlinear systems in the form of

The aim of the robust synchronization is to find an adaptive controller

The hyperchaotic Chen system [

When

In order to observe the synchronization behavior of the hyperchaotic Chen systems, assume that there are two identical hyperchaotic Chen systems where the drive system with subscript 1 drives the response system having identical equations denoted by subscript 2. For system (

Subtract (

The aim of the robust synchronization is to design a controller vector

In this section, depending on the characteristics of disturbance vector

Two robust adaptive techniques are developed based on a control input in the form as

The system external disturbances satisfy

With Assumption

Choose a Lyapunov function as

Besides, the external disturbance

Choosing a smaller

To satisfy the smoothness of controller (

Theorem

In the uncertain response system (

With Assumption

Choose a Lyapunov function as

To verify and demonstrate the effectiveness of the proposed schemes, we will discuss the robust synchronization of the hyperchaotic Chen systems with different external disturbances of

The performance of the proposed robust adaptive synchronization schemes is demonstrated by the following two cases. For this numerical simulations, the system unstructured uncertainties are assumed to be

Choose

The estimated values of the unknown parameters

The states of the hyperchaotic Chen systems (

The states of the hyperchaotic Chen systems (

The states of the error dynamical system (

Take

The estimated values of the unknown parameters

The states of the hyperchaotic Chen systems (

The states of the error dynamical system (

In this paper, the robust synchronization of two hyperchaotic Chen systems with uncertainties and external disturbances is discussed. Depending on the different characteristics of disturbance signals

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Nature Science Foundation of China (no. 11371049) and the Science Foundation of Beijing Jiaotong University (no. 2011JBM130).