Synchronization of a class of fractional-order neural networks with multiple time delays by comparison principles ∗

School of Mathematics and Computational Science, Anqing Normal University, AnQing 246011, China wwzhahu@aliyun.com School of Mathematics, Anhui University, Hefei 230039, China rcwu@ahu.edu.cn School of Mathematics, Southeast University, Nanjing 210096, China jdcao@seu.edu.cn Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia aalsaedi@hotmail.com Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan tahaksag@yahoo.com


Introduction
Neural networks are important nonlinear systems, which have been studied in scientific research and have been applied to the signal processing, parallel computation, optimization, artificial intelligence, and so on.It is known that fractional calculus mainly deals with a generalization of differentiation and integration of arbitrary orders.Compared with classical integer-order models, the fractional-order calculus owns the description of memory and hereditary properties of a variety of processes.Fractional-order models are far better regard the dynamical behaviors of systems [2,6,13,15,16,23,24]. Taking these factors into account, many researchers have incorporated fractional calculus to neural networks and formed fractional-order neural networks [3, 7-9, 14, 25, 27].
Time-delayed models unavoidable exist in biological, engineering systems, and neural networks, see [19-21, 28, 29].Generally, time delays will affect oscillation and instability behavior of a network.Nowadays, the synchronization of fractional-order delayed neural networks has attracted more and more attention.Some synchronization results have been obtained, for instance, in [26], authors studied the finite-time synchronization of fractional-order memristor-based neural networks with time delay.Paper [10] investigated the stability and synchronization of memristor-based fractional-order delayed neural networks.In [4], the synchronization of fractional-order complex-valued neural networks with time delay was studied.Paper [5] discussed the adaptive synchronization of fractional-order memristor-based neural networks with time delay.
However, most existing results related to synchronization of delayed neural networks have been considered with single time delay [17,18].In fact, the differential models with multiple time delays unavoidable exist in neural networks.It is worthy to point out that study about the stability of nonlinear fractional order with multiple time delays seems quite difficulty.There are only few results on the stability and synchronization of fractional-order neural networks with multiple time delays in existing literatures [12,22].Motivated by the above discussions, the aim here is to study the synchronization of fractional-order neural networks with multiple time delays.By constructing a Lyapunov function, applying an inequality of fractional-order and comparison principles of linear fractional equation with multiple time delays, we obtain some sufficient conditions, which can achieve synchronization.The obtained results are novel.
The remainder of this article is organized as follows.In Section 2, some definitions and lemmas are introduced.The model description is also given.In Section 3, the sufficient conditions for synchronization are obtained.Numerical simulations are presented in Section 4. Some conclusions are drawn in Section 5.

Preliminaries and model description
There are some definitions of the fractional-order integrals and derivatives, such as Riemann-Liouville definition and the Caputo definition.From the Laplace transform of fractional derivative, the advantage of the Caputo fractional derivative is that it only requires initial conditions given in terms of integer-order derivatives.Here, the definition of Caputo derivative is adopted.Definition 1. (See [24].)The fractional integral with noninteger order α > 0 for a function x(t) is defined as where t t 0 , t 0 is the initial time, Γ(•) is the gamma function, given by Γ(s) = ∞ 0 t s−1 e −t dt.Definition 2. (See [24].)The Caputo fractional derivative of order α for a function x(t) is defined as We now consider the following fractional-order neural networks with time delays as master system: i, j = 1, 2, . . ., n, where 0 < α < 1, n is the number of units in a neural network, x i (t) denotes the pseudostate variable of the ith unit of master system, c i > 0 is the selfregulating parameters of the ith unit.I i represents the external input of the ith unit, a ij and b ij denote the strength of the jth unit on the ith unit at time t and t − τ j , respectively, τ j > 0 is the transmission delay, f j (x j (t)) and g j (x j (t)) denote the output of the jth unit at time t and t − τ j , respectively.The slave system is given by i, j = 1, 2, . . ., n.Note that y i (t) is the pseudostate vector of slave system and u i (t) is a suitable controller.
To ensure the main results, we present the following assumption and lemmas.
3 Synchronization of fractional-order neural networks with multiple time delays We will discuss the master-slave synchronization of fractional-order neural networks.The purpose is to choose an effective controller to achieve the synchronization of the masterslave systems.Let e i (t) = y i (t) − x i (t), i = 1, 2, . . ., n, be the synchronization errors.
Select the control input function as follows: where k i denotes the controller feedback gain.
Then the error systems are obtained in the form Theorem 1.Under Assumption 1 and since the control function satisfies (5), if the controller feedback gain k i satisfies the inequality then the fractional-order delayed neural networks system (1) synchronizes system (2), where λ = min Nonlinear Anal.Model.Control, 22(5):636-645 Proof.We define the following Lyapunov function: e T (t)e(t), e(t) = (e 1 (t), . . ., e n (t)) T .
From Lemma 1, Note that Substitute these into ( 9), one has Let For convenience, we rewrite (10) as where L = max{b j }.
According to Theorem 1, the following corollary holds.
Corollary 1.Under Assumption 1 and since the control function satisfies (5), if the controller feedback gain k i satisfies the inequality then the fractional-order delayed neural networks system (1) synchronizes system (2).
, the model will reduce to system with single time delay.
Remark 2. Here, we have used comparison principles of fractional-order couple system with multiple time delays.Some synchronization results of fractional-order neural networks with multiple time delays are derived.The method is novel.
Remark 3. Generally, the differential models with multiple time delays unavoidable exist in neural networks.However, the research of synchronization of fractional-order neural networks with multiple time delays has seldom seen.A comparative study reveals that our results are with multiple time delays.Hence, the results obtained in this paper are more general than the existing results dealing with the single time delay [3-5, 10, 12].

Conclusions
Here, we investigated the synchronization of fractional-order neural networks with multiple time delays.By using Lyapunov function and the comparison principles of linear fractional equation with multiple time delays, some sufficient conditions are derived to ensure the synchronization of the master-slave systems.A numerical example is presented to verify the effectiveness of the theoretical results.