^{1}

^{2}

^{3}

^{4}

^{1}

^{1}

^{2}

^{3}

^{4}

This paper is concerned with master-slave synchronization of 4D hyperchaotic Rabinovich systems. Compared with some existing papers, this paper has two contributions. The first contribution is that the nonlinear terms of error systems remained which inherit nonlinear features from master and slave 4D hyperchaotic Rabinovich systems, rather than discarding nonlinear features of original hyperchaotic Rabinovich systems and eliminating those nonlinear terms to derive linear error systems as the control methods in some existing papers. The second contribution is that the synchronization criteria of this paper are global rather than local synchronization results in some existing papers. In addition, those synchronization criteria and control methods for 4D hyperchaotic Rabinovich systems are extended to investigate the synchronization of 3D chaotic Rabinovich systems. The effectiveness of synchronization criteria is illustrated by three simulation examples.

The classic hyperchaotic Rabinovich system was a system of 3D differential equations which was used to describe the plasma oscillation [

There exist various dynamical behaviors of 4D hyperchaotic Rabinovich systems. Synchronization is the typical dynamical behavior of chaotic systems [

In this paper, a master-slave scheme for 4D hyperchaotic Rabinovich systems is constructed. Some global master-slave synchronization criteria for 4D hyperchaotic Rabinovich systems are derived by using the designed controllers. The nonlinear features of error systems remained. Those control methods and synchronization criteria for 4D Rabinovich systems can be used to derive synchronization criteria for 3D Rabinovich systems. Three examples are used to illustrate the effectiveness of our results.

Consider the following 4D Rabinovich system as a master system:

Because the trajectories of a hyperchaotic system are bounded [

One can construct the following slave scheme associated with system (

Let

In this paper, we design

The main purpose of this paper is to design

Now, we give some synchronization results for two 4D hyperchaotic Rabinovich systems described by (

If

One can construct Lyapunov function

It is easy to see that

The inequality described by (

Due to the bound

By virtue of LaSalle Invariant principle, one can derive that the trajectories of (

In [

Rabinovich systems are nonlinear dynamical systems, in which nonlinear terms play an important role in the evolution of trajectories. In [

If

If

If

If

If

If

If

If

Corollary

Consider the following 3D Rabinovich system as a master system:

One can construct the following slave scheme associated with system (

Let

In this paper, we choose

Constructing the Lyapunov function

If

Consider the 4D hyperchaotic Rabinovich system described by (

Then, one can study slave Rabinovich system described by (

If

If

If

The attractor of (

The attractor of (

The trajectories of (

The trajectories of (

The trajectories of (

The trajectories of (

It is easy to see that Corollary

Consider the 4D Rabinovich systems and the error system described by (

The trajectories of (

The trajectories of (

Consider the 3D hyperchaotic Rabinovich systems and the error system described by (

Setting

The trajectories of (

The trajectories of (

We have derived some global synchronization criteria for 4D hyperchaotic Rabinovich systems. We have kept the nonlinear terms of error systems. Those control methods and synchronization criteria for 4D hyperchaotic Rabinovich systems can be used to study the synchronization of 3D chaotic Rabinovich systems. We have used three examples to demonstrate the effectiveness our derived results. In this paper, we only consider the state feedback control. Our future research focus is to design the time-delayed controllers.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This paper is partially supported by the National Natural Science Foundation of China under Grant 61561023, the Key Project of Youth Science Fund of Jiangxi China under Grant 20133ACB21009, the Project of Science and Technology Fund of Jiangxi Education Department of China under Grant GJJ160429, and the Project of Jiangxi E-Commerce High Level Engineering Technology Research Centre.