^{1}

^{2}

^{2}

^{1}

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In this paper, the anti-synchronization of fractional-order chaotic circuit with memristor (FCCM) is investigated via a periodic intermittent control scheme. Based on the principle of periodic intermittent control and the Lyapunov stability theory, a novel criterion is adopted to realize the anti-synchronization of FCCM. Finally, some examples of numerical simulations are exploited to verify the feasibility of theoretical analysis.

Fractional calculus has a history of more than 300 years. It is worth pointing out that fractional-order system has provided infinite memory and more accurately describes natural phenomena than other integer order systems [

The memristor was firstly raised by Chua [

Meanwhile, many scholars investigate the synchronization problems [

Intermittent control, which was first introduced to control linear econometric models in [

Motivated by the above discussions, we propose a periodic intermittent control method for the anti-synchronization of FCCM in this paper. Based on the lyapunov stability theory, a novel and useful criterion of periodic intermittent control is developed by using the differential inequality method. Finally, we have illustrated the effectiveness and feasibility of the proposed approaches by numerical simulations.

This paper is arranged as follows: Section

In this paper, let

The caputo’s fractional derivative for a function

where

Particularly, when

The Mittag–Leffler function of

where

Suppose

For

Let

where

Let

Moreover,

According to the chaotic circuit with memristor [

The chaotic circuit with memristor.

where

Similar to [

Refer to the above model, the fractional-order generalization according to (9) is described as

Usually, in order to obtain the chaotic phenomena, the parameters are selected as ^{
T} to system (10). The simulation results are shown in Figure

The chaotic attractor of FCCM, (a)

Let

where

and

To investigate the anti-synchronization of FCCM, the drive system can be rewritten as

Similarly, the response system with the controller can be described as

where

here

Let

where

In this section, the anti-synchronization problem of FCCM via periodic intermittent control is investigated. First of all, we propose the following assumption.

It can be seen from Figure

Then, we derive the anti-synchronization criteria for the FCCM according to periodic intermittent control scheme and

Suppose Assumption

where

Construct the following Lyapunov function.

Taking the time derivative of

According to the condition (

By Lemma

Similarly, when

According to condition (

From Lemma

From inequality (25) and (28), we summarize that:

When

When

When

When

By induction, when

From Lemma

when

Therefore, from inequality (34) and (35), we have

According to condition (

Hence,

Because

In this section, some numerical simulations are given to illustrate the theoretical analysis.

Based on Figure

by Matlab calculation program, we can get that

Therefore, when we choose

Figure

Dynamical behaviors of the drive system (13) and the response system (14) without periodic intermittent controller (15), where

Dynamical behaviors of the drive system (5) and the response system (6) with periodic intermittent controller (7), where

Synchronization errors between systems (13) and (14) with periodic intermittent controller (15), where

Time evolution of intermittent feedback control gain

In this paper, the anti-synchronization of FCCM via periodic intermittent control has been achieved in finite time based on periodic intermittent control principle and Lyapunov stability theory. In addition, some numerical simulations have been provided to demonstrate the effectiveness of the proposed approach. The result will have potential applications for image encryption, cryptography, and chaotic radar. Our future research is to investigate the anti-synchronization of FCCM with time delay via nonperiodic intermittent control.

All data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interests.

The authors are very grateful to the anonymous reviewers for their valuable comments. This work is supported by the National Science Foundation of China (Grants nos. 51777180, 11771376, 11872327).