The synchronization between fractional-order hyperchaotic systems and integer-order hyperchaotic systems via sliding mode controller is investigated. By designing an active sliding mode controller and choosing proper control parameters, the drive and response systems are synchronized. Synchronization between the fractional-order Chen chaotic system and the integer-order Chen chaotic system and between integer-order hyperchaotic Chen system and fractional-order hyperchaotic Rössler system is used to illustrate the effectiveness of the proposed synchronization approach. Numerical simulations coincide with the theoretical analysis.

During the past decades, fractional calculus has become a powerful tool to describe the dynamics of complex systems such as power systems, mathematics, biology, medicine, secure communication, and chemical reactors [

All of above articles mainly focus on integer-order chaotic systems or fractional-order chaotic systems. There is little information about the synchronization between fractional-order chaotic systems and integer-order chaotic systems [

Motivated by the above discussion, this paper investigates a sliding mode method for synchronization between a class of fractional-order hyperchaotic systems and integer-order hyperchaotic systems. And the integer-order hyperchaotic systems are regarded as response system in the proposed synchronous technique which is simple and theoretically rigorous.

Consider the following fractional-order hyperchaotic system as a drive system

And the response system can be described as

One adds the controller

Let the controller

From (

In accordance with the active control design procedure, the nonlinear part of the error dynamics is eliminated by the following choice of the input vector [

The error system (

To design a sliding mode controller, one has two steps. First, one constructs a sliding surface that represents a desired system dynamics. Next, one develops a switching control law such that a sliding mode exists on every point of the sliding surface, and any states outside the surface are driven to reach the surface in a finite time [

In the sliding mode, the sliding surface and its derivative must satisfy

Consider

One can get that

Replacing for

To satisfy the sliding condition, the discontinuous reaching law is chosen as follows:

In the sliding phase, it implies that

Now, the total control law can be defined as follows:

Replacing

The following system is as follows:

Consider a system given by the following linear state space form with inner dimension

According to Theorem

This section presents two illustrative examples to verify and demonstrate the effectiveness of the proposed control scheme. Case

Synchronization between fractional-order and integer-order hyperchaotic Chen systems.

Consider Chen hyperchaotic system which is written as [

When

Take the fractional-order system with fractional-order

The controller parameters are chosen as

The simulation results are given in Figure

Synchronization errors between Chen systems.

Synchronization between integer-order hyperchaotic Chen system and fractional-order hyperchaotic Rössler system.

Consider hyperchaotic Rössler system which is written as [

Similarly, take the fractional-order Rössler hyperchaotic system with fractional-order

We choose the design parameters in the simulations as

The synchronization errors are shown in Figure

Synchronization between integer-order Chen system and fractional-order Rössler system.

In this paper, the problem of synchronization between fractional-order hyperchaotic systems and integer-order hyperchaotic systems is investigated. The integer-order hyperchaotic system is regarded as the response system. A sliding mode controller is designed to synchronize two systems with different orders successfully. It is rigorously proven that the proposed synchronization approach can be achieved between two different order hyperchaotic systems. Some numerical simulations are presented to show the applicability and feasibility of the proposed scheme.

This wok was supported by the “111” Project from the Ministry of Education of People’s Republic of China and the State Administration of Foreign Experts Affairs of People’s Republic of China (B12007) and the National Science and Technology Supporting Plan from the Ministry of Science and Technology of People’s Republic of China (2011BAD29B08).