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Chaotic systems are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems with Gaussian fluctuation. A fractional integral (FI) sliding surface is proposed for synchronizing the uncertain fractional-order system, and then the sliding mode control technique is carried out to realize the synchronization of the given systems. One theorem about sliding mode controller is presented to prove that the proposed controller can make the system achieve synchronization. As a case study, the presented method is applied to the fractional-order Chen-Lü system, and simulation results show that the proposed control approach is capable to go against Gaussian noise well.

Synchronization, which means “things occur at the same time or operate in unison,” has received a great deal of interest among scientists from various fields in the last few years, especially in fractional-order chaotic systems [

A basic configuration for chaos synchronization is the drive-response pattern, where the response of chaotic system must track the drive chaotic trajectory. Some approaches based on this configuration have been attained to achieve chaos synchronization in fractional-order chaotic systems, such as Pecora and Carroll (PC) control [

All of the methods mentioned above have been used to synchronize the deterministic fractional-order chaotic systems. However, noise-induced synchronization in chaotic systems is a practical phenomenon due to the fact that noises are ubiquitous in natural and synthetic systems, and up till now, it has been studied by many investigators from different areas [

This paper is organized as follows. In Section

Consider the following class of fractional-order chaotic system excited by Gaussian white noise, which is described by

where

with

Let system (

where

where

Thus, the control problem considered in this study is that, for chaotic driving system (

where

In the following context, we will design sliding mode controller to establish synchronization between driving system (

Now, the control input vector

where

in which

So the error system (

Here, a new fractional integral (FI) switching surface is given as follows:

where

As we all know, when the system is controllable in the sliding mode, the switching surface should satisfy the following conditions:

together with

Substituting (

Therefore, the equivalent control law is obtained by

In real-world applications, the Gaussian white noise

To design the sliding mode controller, we consider the constant plus proportional rate reaching law [

where

So, we can get the controller

Further, according to the control law and the updated law, the controller is given by

And the error system can be rewritten in the following differential form:

If the controller is selected as (

Consider a Lyapunov function constructed by the mean square norm of

According to the definition of derivative, it is found that

While, from (

Substituting (

Taking expectations to (

Therefore

Equation (

This completes the proof.

In this part, to confirm the validity of proposed method, we numerically examine the synchronization between fractional-order Chen system [

Here, we assume that the Chen system drives the Lü system. Hence, the driving system (fractional-order Lü system) is described as

which can also be written as

in which,

It has been shown that the fractional-order Lü system can demonstrate chaotic behavior [

The response system (fractional-order Chen system) is given as follows:

which can be written in the following form:

where

in which system will exhibit chaotic behavior [

where

Now, we apply the proposed sliding control approach to finish synchronization between fractional-order Lü system driven by Gaussian white noise and fractional-order Chen system. Here, we define the error states as

and the sliding mode surface as

The control law is given by

In the numerical simulations, the initial conditions are set as

(a) The state trajectories of the system (

(a) The time evolution of synchronization error

Mean value of synchronization errors

Variance of the state estimation errors

From the simulation results, it can be concluded that the obtained theoretic results are efficient and feasible for synchronizing fractional uncertain dynamical systems, and the proposed controller guarantees the convergence of the error system.

In this paper, we focus on the problem of synchronization between fractional-order chaotic systems with Gaussian fluctuation by the method of fractional-order sliding mode control. A fractional sliding surface is introduced, and the sliding mode controller is proposed for synchronization. Furthermore, convergence property has been analyzed for the error dynamics after adding proposed controllers. It has been shown that the fractional-order chaotic systems under uncertain environment can achieve synchronization by proper choice of control parameters (

This work was supported by the NSF of China (Grant nos. 11372247 and 11102157), Program for NCET, the Shaanxi Project for Young New Star in Science & Technology, NPU Foundation for Fundamental Research, and SRF for ROCS, SEM.