Considering the fact that the exponential synchronization of neural networks has been widely used in theoretical research and practical application of many scientific fields, and there are a few researches about the exponential synchronization of fractional-order memristor-based neural networks (FMNN). This paper concentrates on the FMNN with time-varying delays and investigates its exponential synchronization. A simple linear error feedback controller is applied to compel the response system to synchronize with the drive system. Combining the theories of differential inclusions and set valued maps, a new sufficient condition concerning exponential synchronization is obtained based on comparison principle rather than the traditional Lyapunov theory. The obtained results extend exponential synchronization of integer-order system to fractional-order memristor-based neural networks with time-varying delays. Finally, some numerical examples are used to demonstrate the effectiveness and correctness of the main results.

Chua already supposed the existence of memristor in 1971 [

In recent years, more and more attentions have been put on the dynamical analysis of memristor-based neural networks, such as the investigation of stability [

Moreover, the fractional-order models can better describe the memory and genetic properties of various materials and process, so the fractional-order models have received a lot of research attentions than integer-order models. In recent years, with the improvement of fractional-order differential calculus and fractional-order differential equations, it is easy to model and analyze practical problems [

On the other hand, the stability and synchronization of FMNN without time delay have been deeply studied such as in [

Motivated by the above discussion, this paper studies the exponential synchronization of FMNN with time-varying delays. The main contributions of this paper can be listed as follow.

The rest of this paper is organized as follows. Preliminaries including the introduction of Caputo fractional-order derivative, model description, assumptions, definitions and lemmas are presented in Section 2. Section 3 introduces the sufficient condition for exponential synchronization of the FMNN. In Section4, the numerical simulations are presented. Section5 gives the conclusion of this paper.

Compared to the integer-order derivatives, we know the distinct advantage of Caputo derivative is that it only requires initial conditions from the Laplace transform of fractional derivative, and it can represent well-understood features of physical situations and making it more applicable to real world problems [

Particularly, when 0<

In this paper, referring to some relevant works on FMNN [_{i}_{i}_{i}_{j}_{j},g_{j}_{ij}_{j}_{ij}_{j}_{j}(_{ij}_{ij}_{ij}_{ij}_{ij}_{i}_{i}_{i}_{ij}_{i}_{i}_{i}_{i}_{i}

In the rest of paper, we first make following assumption for system

Assumption1: For _{1}, _{2} ∈ _{j}_{j}_{i}_{j}_{1} ≠ _{2} and _{j}_{j}

We consider system

Where
_{i}^{t}) is a liner error feedback control function which defined by _{i}^{t}) = _{i}_{i}_{i}_{i}_{1}(_{2}(_{n}(^{T}, where _{i}(_{i}_{i}_{ij}_{j}_{ij}_{j}_{j}(_{ij}_{j}_{ij}_{j}_{j}(_{i}^{t}) = _{i}_{i}_{i}_{i}e_{i}_{i}

According to the theories of differential inclusions and set valued maps [_{i}_{i}

And

Where

And

where

_{i}_{i}_{1}(_{2}(_{n}(^{T} of error system _{0} − _{0}],^{n}_{i}

(i) _{ij}_{j}_{j}_{j}_{ij}_{j}_{j}_{j}_{ij}F_{j}_{j}

(ii) _{ij}_{j}_{j}(_{j}_{j}_{j}(_{ij}_{j}_{j}(_{j}_{j}_{j}(_{ij}G_{j}_{j}_{j}(

where

_{i}_{i}

For _{i}_{i}

For _{i}_{i}

For _{i}_{i}_{i}_{i}

Then complete the proof of part (i). In the similar way, part(ii) can be easily hold.

We present the exponential stability results for the synchronization error system of FMNN when the error system

_{1},_{2},…,_{n}

_{i}_{i}_{i}

Evaluating the fractional order derivative of _{i}

Define

We will prove that _{0}>0. Otherwise, since _{0} − _{0}], there must exist _{1} ≥ _{0} and some

Moreover, from inequality

Therefore

It shows

This completes the proof.

In this section, we will give two numerical examples to demonstrate our analysis on exponential synchronization of FMNN.

_{1} = _{2} = 1, _{11}(_{1}(_{22}(_{2}(_{j}^{t}^{t}_{1},_{2})^{T} = (0,0)^{T}, _{i}_{i}_{i}_{i}_{i}_{i}_{i}^{T} which can be seen in Figure

We consider system _{i}_{i}_{i}_{i}_{i}_{1} = −9.5, _{2} = −10.5. From Theorem1, when we take _{j}_{1} = _{2} = _{1} = _{2} = _{1}= _{2} = 0.1, we can easily know _{1} = −9.5, _{2} = −10.5, we can get is true when _{1} < −1.703, _{2} < −0.232. So when

It satisfies the condition of Theorem 1, then the exponential synchronization of drive-response system is achieved.

When the response system with this controller, we get state trajectories of variable _{1}(_{1}(_{2}(_{2}(_{1}(_{2}(_{1}(_{1}(_{2}(_{2}(_{1}(_{2}(

The chaotic attractors of fractional-order memristor-based neural networks

Exponential synchronization of state variable with cntroller (_{1}(_{1}(_{2}(_{2}(

Synchronization error between the drive and response system (_{1}(_{2}(

_{1} = _{2} = _{3} = 1,

And _{j}^{t}^{t}_{1},_{2},_{3})^{T} = (0,0,0)^{T} _{i}_{i}_{i}_{i}_{i}_{i}^{t}) = _{i}_{i}_{i}_{i}_{1} = −9.5, _{2} = −10.5, _{3} = −11. From Theorem1, we take _{j}(_{1} = _{2} = 0.1 _{1} = _{2} = _{1} = _{2} = 0.1. According to
_{ij}_{ij}_{i}_{1} = −9.5, _{2} = −10.5, _{3} = −11 we can get

It suggests the condition of Theorem 1 is satisfied, then drive-response system achieves the synchronization.

When the response system with this controller, we get state trajectories of variable _{1}(_{1}(_{2}(_{2}(_{3}(_{3}(_{1}(_{2}(_{3}(_{1}(_{1}(_{2}(_{2}(_{3}(_{3}(_{1}(_{2}(_{3}(

In addition, we choose _{1} = −9.5, _{2} = −10.5, _{3} = −11, according to the Theorem1, it needs the following inequalities to hold:

So, we just need

Synchronization of state variable with controller (_{1}(_{1}(_{2}(_{2}(_{3}(_{3}(

Synchronization error between the drive and response system (_{1}(_{2}(_{3}(

The relation of time-varying delay

This paper achieves the exponential synchronization of a class of FMNN with time-varying delays by using linear error feedback controller. Based on comparison principle, the new theorem is derived to guarantee the exponential synchronization between the drive system and response system. The methods proposed for synchronization is effective and it is easy to achieve than other complex control methods. Moreover, it can be extended to investigate other dynamical behaviors of fractional-order memristive neural networks, such as realizing the lag synchronization or anti-synchronizaton of this system based on the suitable controller. These issues will be the topic of future research. Finally, numerical examples are given to illustrate the effectiveness of the proposed theory.