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A modified function projective synchronization for fractional-order chaotic system, called compound generalized function projective synchronization (CGFPS), is proposed theoretically in this paper. There are one scaling-drive system, more than one base-drive system, and one response system in the scheme of CGFPS, and the scaling function matrices come from multidrive systems. The proposed CGFPS technique is based on the stability theory of fractional-order system. Moreover, we achieve the CGFPS between three-driver chaotic systems, that is, the fractional-order Arneodo chaotic system, the fractional-order Chen chaotic system, and the fractional-order Lu chaotic system, and one response chaotic system, that is, the fractional-order Lorenz chaotic system. Numerical experiments are demonstrated to verify the effectiveness of the CGFPS scheme.

Chaotic phenomenon in nonlinear system has attracted more and more attentions in recent years [

Up to now, various types of chaos synchronization have been proposed, such as complete synchronization (CS) [

However, the response system could be synchronized to the drive system with a function matrix in FPS scheme, and there are only one drive system and one response system. Therefore, the FPS between multidrive systems and one response system, where the scaling function matrix comes from multidrive systems, is interesting and general topic. It is obvious that multidrive systems and one response system synchronization in FPS scheme can additionally enhance the security of communication; this is due to the fact that the transmitted signals can be split into several parts, and each part may be loaded in different drive systems; or the transmitted signals can be divided in time into different intervals, and the signals in different intervals may be loaded in different drive systems [

The layout of this paper is organized as follows. In Section

In this section, we will demonstrate the CGFPS scheme for fractional-order chaotic systems. As is well known, there are several definitions about fractional derivative. This paper will use the Caputo definition of the fractional derivative, described as follows [

Consider the fractional-order multidrive chaotic systems and one fractional-order response chaotic system described as follows:

Before we give a detailed definition of the CGFPS scheme, let us first recall the definitions of the FPS scheme and the compound synchronization scheme.

For the scaling-drive system (

For the scaling-drive system (

Now, we establish the CGFPS scheme for fractional-order chaotic systems.

For the scaling-drive system (

If

The scaling function matrix

The scaling function matrix

The scaling-drive system (

Next, we will discuss how to achieve CGFPS between the drive systems (

Now, the controller

Using the above controller (

For response system (

Combining

Therefore, we can yield the following fractional-order system about errors:

Here, let us recall one result for the stability of nonlinear fractional-order system [

If one chooses the feedback controller

Use

According to the stability for nonlinear fractional-order systems [

This result implies that the compound generalized function projective synchronization (CGFPS) between drive systems (

In this section, numerical experiments are used to verify the effectiveness of the CGFPS scheme in our paper. Some examples are given and the numerical simulations are performed.

Now, choose the fractional-order Arneodo chaotic system as the scaling-drive system, which is given by [

The chaotic attractor of system (

Then, choose the fractional-order Chen chaotic system and the fractional-order Lu chaotic system as the base-drive systems. The fractional-order Chen chaotic system [

The chaotic attractor of system (

The chaotic attractor of system (

Next, we choose the fractional-order Loren chaotic system as response system, which is given by [

The chaotic attractor of system (

Next, we discuss how to realize the CGFPS between drive systems (

According to Section

If we choose the feedback controller

Equations (

Now, two cases with numerical simulations are given to verify the effectiveness of proposed scheme.

Choose

The CGFPS errors between the multidrive systems (

Choose

The CGFPS errors between the multidrive systems (

In this paper, we present a new synchronization scheme for fractional-order chaotic system and called this type of synchronization compound generalized function projective synchronization, or briefly denoted it by CGFPS. In this scheme, there are one scaling-drive system, more than one base-drive system, and one response system, and the scaling function matrix comes from multidrive systems. So, the CGFPS in this paper is different from all the previous synchronization reported before. To verify its effectiveness, we achieve the CGFPS between three drive systems and one response system, where the scaling-drive system is the fractional-order Arneodo chaotic system, the base-drive systems are the fractional-order Chen chaotic system and the fractional-order Lu chaotic system, and the response system is the fractional-order Loren chaotic system. Numerical simulations suggest that the presented CGFPS scheme works well. It is worth mentioning that the CGFPS scheme in our paper can also be used for other fractional-order chaotic systems or integer order chaotic systems.

The authors declare no conflict of interests.

Ping Zhou and Chunde Yang proposed and designed the research. Ping Zhou, Hao Cai, and Chunde Yang performed the simulations. Hao Cai and Ping Zhou analyzed the simulation results. Ping Zhou and Chunde Yang wrote the paper. All authors have read and approved the final paper.