Finite time impulsive synchronization of fractional order memristive BAM neural networks

Finite time impulsive synchronization of fractional order memristive BAM neural networks

Communicated by Dr. Jin-Liang Wang

Journal Pre-proof

Finite time impulsive synchronization of fractional order memristive BAM neural networks Lingzhong Zhang, Yongqing Yang PII: DOI: Reference:

S0925-2312(19)31765-5 https://doi.org/10.1016/j.neucom.2019.12.056 NEUCOM 21685

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

10 May 2019 24 October 2019 15 December 2019

Please cite this article as: Lingzhong Zhang, Yongqing Yang, Finite time impulsive synchronization of fractional order memristive BAM neural networks, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.12.056

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Finite time impulsive synchronization of fractional order memristive BAM neural networks

∗

Lingzhong Zhanga,b†, Yongqing Yangb a.School of Electrical Engineering and Automation, Changshu Institute of Technology, Changshu 215500, Jiangsu, PR China b.School of Science, Jiangnan University, Wuxi 214122, PR China

Abstract This paper is concerned with the finite time impulsive synchronization problem of fractional order memristive BAM neural networks(MBAMNNs) with switching jumps mismatch. By utilizing the properties of fractional calculus and comparison principle, a novel fractional order result about finite time impulsive stability is obtained. In the light of the double layer structure of fractional order MBAMNNs, two impulsive controllers are designed for the response fractional order MBAMNNs. By combing properties of Gamma functions and some analysis techniques of impulsive interval, several sufficient conditions are given which ensure finite time synchronization of fractional order MBAMNNs, and the upper bound of the setting time for synchronization is estimated, which is related to the fractional order of the system. Finally, numerical simulation is provided to demonstrate the effectiveness of the obtained results. Keywords: memristor; impulsive control; finite time synchronization; BAM neural networks;

1

Introduction

As a typical dynamic process, finite synchronization has attracted much attention due to its optimality in convergence time and its wide occurrence in satellite formation and load cooperative transportation. In ∗

This work was jointly supported by the Natural Science Foundation of Jiangsu Province No.BK20181033

†

E-mail: [email protected]

1

reality, network systems might often be expected to achieve stability or synchronization as soon as possible, especially in engineering fields. In order to obtain faster convergence rate in network systems, an effective method is to use finite time synchronization control techniques. In recent years, more and more effective control strategies have been designed to achieve system stability or synchronization in finite time[1]-[4]. Bidirectional associative memory(BAM) neural networks was originally proposed by Kosko[5], and has been widely applied in pattern recognition and artificial intelligence because they extend the singe layer auto associative Hebbian circuit to a two layer patter matched hetero associative correlation[6]. The two layer patter matched hetero associative circuit can help networks store and recall patter pairs. Consequently, the reverberation interference information between two layer circuit would be inevitably related to information which is derived by decoding in BAM. In this way, there are some hereditary properties in the process of intelligent information processing. Based on this fact, it is necessary to take into account merging infinite memory into this model[7]. BAM neural network can be implemented in a circuit where the common connection weights and the self-feedback connection weights are established by memristor[26]. The memristor which is well known as the fourth fundamental nonlinear circuit element was proposed by Chua[8, 9, 10]. Memristor owns performances and memory more like biological synapses than the resistor[11, 12], and possesses the characteristic of pinched hysteresis[13]. Based on these characteristics, some researchers founded a new class of BAM neural networks named memristive BAM neural networks(MBAMNNs) by replacing resistor with memristor and studied the dynamics behaviors of MBAMNNs to realize its better application in patter recognition and associative memories in the future[14, 15]. Fractional order calculus is a generalization of the usual derivation and integration to arbitrary noninteger order[16]. Fractional derivatives accumulate global information of functions in weighted form, which can describe the memory properties of neurons[17]-[20]. Many physical systems, such as vehicles moving on top of sand or muddy road are inaccurate to be described by classical integer-order ones[19]. Compared with integer-order differential systems, fractional order differential system has infinite memory and genetic[20].

The combination of fractional order calculus and the memristive

neural network can generate more dynamic characteristics. Fractional order memristive network systems have appeared naturally in a number of fields, such as physics, engineering, biophysics,

2

blood flow phenomena[21]. Researchers have introduced fractional differential operators into MBAMNNs, and established a model of fractional order MBAMNNs, which is superior to traditional neural networks in improving estimation accuracy and parameter identification[21]-[23]. In 2014, Professor Shen proposed that there is no equilibrium point of finite time stability for fractional order systems[24]. Based on this work, it can be determined that fractional order systems can not achieve finite time stability under continuous control. However, through numerical simulation experiments, it is found that the fractional order system can achieve finite time stability of equilibrium point by using discontinuous control method. Recent years have witnessed an increasing interest in finite synchronization of fractional order systems from science and engineering fields and many better results have been proposed in the literature[25]-[30]. Much effort has been devoted to design proper control methods to regulate fractional order systems into a synchronous motion in finite time, such as linear feedback control[26], adaptive control[27], and hybrid feedback control[28]. In [29], based on generalized Gronwall inequality and Laplace transform, authors obtained finite time synchronization for fractional order neural networks(FNNs) by designing linear feedback control. The finite time projective synchronization was studied by means of Gromwall-Bellman integral inequality in [30]. As an effective strategy, impulsive control allows systems to obtain discontinuous control inputs, when system does not need continuous control input, or can not provide continuous control input, impulse control method can be used[31]. However, due to the lack of fractional order theory, there are few published papers considering impulse control in finite time synchronization of fractional order systems. On the other hand, in [29, 30], the estimation of setting time is not given, the relationship between fractional order and synchronization setting time is not discussed. Therefore, how to design the impulsive controller and a comprehensive analysis of finite time synchronization in fractional order MBAMNNs is in great demand. The main results of this paper contribute to the development of impulsive controller to realize finite time synchronization of fractional order MBAMNNs, include (1) We present a new conclusion on the finite time stability criterion of fractional impulsive MBAMNNs by using the fractional order comparison theory and properties of fractional calculus. (2) By combining some analysis techniques, fractional order inequality and Lyapunov theory, the finite time synchronization criteria of fractional order MBAMNNs are established by designing effective controller. (3) The upper bound estimation of setting time is given, and the influence

3

of order on setting time is analyzed. (4) The relationship between control parameters and synchronization domain is discussed in detail. In addition, we have weakened the restrictions of the driving function, such as fj (±Tej ) = fj (±Tj ) = 0, f (0) = 0, (Tj , Tej are switching jumps)[35, 36, 37]. These restrictions conditions are

not necessary here.

The organization of this paper is as follows. In section 2, some important preliminaries and fractional order MBAMNNs model formulation are presented. In section 3, some criteria for finite time impulsive synchronization of fractional order MBAMNNs are obtained. In section 4, numerical example is given to illustrate the theoretical results. Finally, a conclusion is given.

2

Preliminaries

This section, some basic definitions concerning fractional calculation are introduced, and some Lemmas and problem formulation are presented, which are helpful for the following sections.

2.1

Caputo fractional-order operator

Definition 1. [33]The fractional order integral of order α for a function f (t) is defined as 1 Γ(α)

Itα0 f (t) = where α > 0, t > t0 and Γ(α) =

R∞ 0

Z

t

t0

(t − s)α−1 f (s)ds,

tα−1 e−t dt.

There are some definitions for fractional derivative, such as Riemann-Liouville fractional derivative, Caputo fractional derivative. Since the initial conditions for Caputo fractional differential equation has the same form as traditional differential equation, which has well understood physical meaning and simple Laplace transform. Therefore, Caputo fractional derivative will be main tool in this paper. Definition 2. [33] The Caputo fractional order derivative of order α for a function f is defined as α C Dt0 ,t f (t)

=

1 Γ(n − α)

Z

t

t0

(t − s)(n−α−1) (

d n ) f (s)ds, ds

where t > t0 , α is the order of derivation, n − 1 < α < n, n ∈ Z + . Especially, when 0 < α < 1, α C D0,t f (t) =

1 Γ(1 − α)

4

Z

t

t0

0

(t − s)−α f (s)ds.

For simplicity, denote Dα f (t) as the

3

α C D0,t f (t).

System description

The dynamics of the ith neuron of fractional order MBAMNNs with switching jumps mismatch is described by

 m m X X  α  D x (t) = −µ (x (t))x (t) + a (x (t))f (y (t)) + bji (xi (t))fj (yj (t − τj )) + Ii ,  i i i i ji i j j   j=1

j=1

n n  X X   α  cij (yj (t))gi (xi (t)) + dij (yj (t))gi (xi (t − τi )) + Jj ,  D yj (t) = −νj (yj (t))yj (t) + i=1

(1)

i=1

where i = 1, 2, · · ·, n; j = 1, 2, · · ·, m; µi (·) > 0 and νj (·) > 0 denote the rates of neuron self inhibition; xi (t) and yj (t) denote the voltages of capacitors Cxi and Cyj , respectively; fj (·) and gj (·) are the activation functions; Ii , Jj represent external input, respectively; τj and τi correspond to the delays and satisfied 0 6 τj 6 τ , 0 6 τi 6 τ (τ is a constant); aji (xi (t)), bji (xi (t)), cij (yj (t)) and dij (yj (t)) are neural connection memristive weights matrices, defined as µi (xi (t)) =

νj (yj (t)) =

m ? Wji 1 Wji 1 X ? { (Wji + Wji )sgnji + }, aji (xi (t)) = × sgnji , bji (xi (t)) = × sgnji , Ci j=1 Ri Ci Ci n 1 X f 1 ? fji { (Wji + W )sgnji + }, cij (yj (t)) = e e Ci i=1 Ri

fij W × sgnij , dij (yj (t)) = ei C

f? W ji × sgnij , e Ci

? f f ? denote the memductances of memristors when i = j, sgnji = −1, otherwise i 6= j, sgnji = 1. Wji , Wji , Wji , W ji

fgji , M f? . Mf ij shows the memristor between xi (t) and fj (yj (t)), M ? shows the memristor Mf ij , Mf?ij , M gji f ij

fgji shows the memristor between yj (t) and gi (xi (t)), M f? shows the between xi (t) and fj (yj (t − τj )), M gji

ej represent the parallel resistors. Chua[10] has proved memristor between yj (t) and gi (xi (t − τi )). Ri and R that memristor exhibit two sufficient equilibrium states. Fig.1 shows the typical feature of memductance of the memristor. According  to the Fig.1. and the properties    ?    µi , |σ| 6 Ti ,  a?ji , µi (σ) = aji (σ) =      µ?? , |σ| > Ti ,  a?? , i ji      ν ? , |σ| 6 Tej ,  c? ,   j ij νj (σ) = cij (σ) =      ν ?? , |σ| > Tej ,  c?? , j

ij

of memristor, for i  = 1, 2, · · ·, n, j = 1, 2, · · ·, m, we set    b?ji , |σ| 6 Ti , |σ| 6 Ti , bji (σ) =    b?? , |σ| > Ti , |σ| > Ti , ji    b? , |σ| 6 Tei ,  |σ| 6 Tei , ij dij (σ) =    b?? , |σ| > Tei , |σ| > Tei , ij

? ?? ? ?? ? ?? ? ?? ? where the switching jumps Ti > 0 and Tej > 0 are constant, µ?i , µ?? i , aji , aji , bji , bji , νj , νj , cij , cij , dij and

d?? ij are all constants numbers.

5

Figure 1: Figure of current-voltage characteristics of the memoristor Let x(t) = (x1 (t), x2 (t), · · ·, xn (t))T , y(t) = (y1 (t), y2 (t), · · ·, ym (t))T , the initial conditions of fractional order MBAMNNs (1) is x(s) = φ1 (s) ∈ C([−τ, 0], Rn ) and y(s) = φ2 (s) ∈ C([−τ, 0], Rm ). System (1) is referred as drive system, the corresponding response system is given by  m m X X  α  D x ¯ (t) = −µ (¯ x (t))¯ x (t) + a (¯ x (t))f (¯ y (t)) + bji (¯ xi (t))fj (¯ yj (t − τj )) + Ii + ui (t),  i i i i ji i j j   j=1

j=1

n n  X X   α  D y ¯ (t) = −ν (¯ y (t))¯ y (t) + c (¯ y (t))g (¯ x (t)) + dij (¯ yj (t))gi (¯ xi (t − τi )) + Jj + vj (t), j j j j ij j i i  i=1

(2)

i=1

where ui (t) and vj (t) are controllers to be designed later. Let x ¯(t) = (¯ x1 (t), x ¯2 (t), · · ·, x ¯n (t))T , y¯(t) = (¯ y1 (t), y¯2 (t), ···, y¯m (t))T , the initial conditions of fractional order MBAMNNs (2) is x ¯(s) = ϕ1 (s) ∈ C([−τ, 0], Rn ) and y¯(s) = ϕ2 (s) ∈ C([−τ, 0], Rm ). Systems (1) and (2) are discontinuous switched systems, and the solutions of systems (1) and (2) are studied in Filippov regularization sense[32]. In the following, fractional order Filippov solution are introduced. Definition 3. For system

    Dα u(t) = g(t, u), t > 0, u ∈ Rn ,    u(0) = u0 ,

(3)

where g(t, u) is discontinuous in u. A set-valued map G : Rn → Rn is defined as G(t, u) =

\

\

co[g(t, B(u, δ)/N )],

δ>0 µ(N )=0

where co[g(t, B(u, δ)/N )] is the closure of the convex hull of set E, E ⊂ Rn . B(u, δ) = {v : kv − uk 6 δ} and µ(N ) is the Lebesgue measure of set N . A vector function u(t) defined on a nondegenerate interval I ⊂ R

6

is called a Filippov solution of system (3), if it is absolutely continuous on any subinterval [t1 , t2 ] of I, for a.a.t ∈ I, u(t) satisfies the differential inclusion Dα u(t) ∈ G(t, u). Based on Definition 3, the systems (1) and (2) can be written as follows  m m X X  α  D x (t) ∈ −co[µ (x (t))]x (t) + co[a (x (t))]f (y (t)) + co[bji (xi (t))]fj (yj (t − τj )) + Ii ,  i i i i ji i j j   j=1

j=1

n n  X X   α  D y (t) ∈ −co[ν (y (t))]y (t) + co[c (y (t))]g (x (t)) + co[dij (yj (t))]gi (xi (t − τi )) + Jj , j j j j ij j i i  i=1

and

(4)

i=1

 m m X X   α  co[bji (¯ xi (t))]fj (¯ yj (t − τj )) + Ii co[a (¯ x (t))]f (¯ y (t)) + D x ¯ (t) ∈ −co[µ (¯ x (t))]¯ x (t) +  ji i j j i i i i    j=1 j=1         +ui (t), n n  X X    Dα y¯j (t) ∈ −co[νj (¯ yj (t))]¯ yj (t) + co[cij (¯ yj (t))]gi (¯ xi (t)) + co[dij (¯ yj (t))]gi (¯ xi (t − τi )) + Jj     i=1 i=1        +vj (t),

(5)

where          ? ?    b?ji , |σ| < Ti , aji , |σ| < Ti , µi , |σ| < Ti ,                co[µi (σ)] = µ?? |σ| > Ti , |σ| > Ti , co[bji (σ)] =  b?? |σ| > Ti , co[aji (σ)] =  a?? ji , ji , i ,                     co{b?ji , b??  co{a?ji , a??  co{µ?i , µ?? ji }, |σ| = Ti , ji }, |σ| = Ti , i }, |σ| = Ti ,          ?    d?ij , |σ| < Tej , c?ij , |σ| < Tej , |σ| < Tej ,    νj ,             e ej , co[cij (σ)] = c?? co[d (σ)] = co[νj (σ)] = νj?? , ij d?? |σ| > Tej , , |σ| > T , |σ| > T j ij , ij                      e e  co{d?ij , d??  co{c?ij , c??  co{νj? , νj?? }, |σ| = Tej , ij }, |σ| = Tj , ij }, |σ| = Tj ,

? ?? co{µ?i , µ?? ¯ji ], aji = min{a?ji , a?? ¯i ], µi = min{µ?i , µ?? ¯i = max{µ?i , µ?? ¯ji = i } = [µi , µ i }, µ i }, co{aji , aji } = [aji , a ji }, a ? ?? ? ?? ? ?? ¯ ? ?? ¯ max{a?ji , a?? ji }, aji = max{|aji |, |aji |}, co{bji , bji } = [bji , bji ], bji = min{bji , bji }, bij = max{bji , bji }, bji = ? ?? max{|b?ji |, |b?? ¯j ], ν j = min{νj? , νj?? }, ν¯j = max{νj? , νj?? }, co{c?ij , c?? ¯ij ], cij = ji |}, co{νj , νj } = [ν i , ν ij } = [cij , c ? ?? ? ?? ? ?? ¯ ¯ min{c?ij , c?? ¯ij = max{c?ij , c?? ij }, c ij }, cij = max{|cij |, |cij |}, co{dij , dij } = [dij , dij ], dij = min{dij , dij }, dij = ? ?? max{d?ij , d?? ij }, dij = max{|dij |, |dij |}.

Or there exist µ ´i ∈ co[µi (xi (t))], a ´ji ∈ co[aji (xi (t))], ´bji ∈ co[bji (xi (t))], ν´j ∈ co[νj (xj (t))], c´ij ∈ co[cij (xj (t))], d´ij ∈ co[dij (xj (t))]. µ `i ∈ co[µi (¯ xi (t))], a `ji ∈ co[aji (¯ xi (t))], `bji ∈ co[bji (¯ xi (t))], ν`j ∈ co[νj (¯ xj (t))],

7

c`ij ∈ co[cij (¯ xj (t))], d`ij ∈ co[dij (¯ xj (t))], such that

 m m X X  α ´bji fj (yj (t − τj )) + Ii ,  D x (t) = −´ µ x (t) + a ´ f (y (t)) +  i i i ji j j   j=1

j=1

(6)

n n  X X   α  D y (t) = −´ ν y (t) + c ´ g (x (t)) + d´ij gi (xi (t − τi )) + Jj , j j j ij i i  i=1

and

i=1

 m m X X  α `bji fj (¯  D x ¯ (t) = −` µ x ¯ (t) + a ` f (¯ y (t)) + yj (t − τj )) + Ii + ui (t),  i i i ji j j   j=1

j=1

(7)

n n  X X   α  D y ¯ (t) = −` ν y ¯ (t) + c ` g (¯ x (t)) + d`ij gi (¯ xi (t − τi )) + Jj + vj (t), j j j ij i i  i=1

i=1

The appropriate impulsive controllers ui (t), vj (t) are designed as

 0    ui (t) = −kix exi (t) − sgn(exi (t))(ξix |exi (t)|α + η x ), t 6= tk

(8)

   vj (t) = −k y ey (t) − sgn(ey (t))(ξ y |ey (t)|α0 + η y ), t 6= tk j j j j j

where k ∈ N+ , exi (t) = x ¯i (t) − xi (t) and eyj (t) = y¯j (t) − yj (t) are synchronization errors. kix , kjy , ξix , ξjy , η x , η y are constants to be determined. α0 is a constant, 0 < α0 < α. The time sequence {tk , k ∈ N+ } is a strictly x increasing sequence of impulsive moment and assumed that limk→∞ tk = +∞, exi (tk ) = exi (t− k ) = limt→t− e (t), k

eyj (tk )

=

eyj (t− k)

= limt→t− ey (t), then the synchronization error systems can be expressed as k

 m X  α x   D e (t) = −` µ x ¯ (t) + µ ´ x (t) + {` aji fj (¯ yj (t)) − a ´ji fj (yj (t))} i i i i  i    j=1    m  X    + {`bji fj (¯ yj (t − τj )) − ´bji fj (yj (t − τj ))} − kix exi (t) − η x sgn(exi (t)),     j=1        x − x x −  ∆exi (tk ) = exi (t+ k ) − ei (tk ) = µ ei (tk ), i = 1, 2, · · ·, n; k = 1, 2, · · · n  X   α y  D e (t) = −` ν y ¯ (t) + ν ´ y (t) + {` cij gi (¯ xi (t)) − c´ij gi (xi (t))}  j j j j j    i=1    n  X    + d`ij gi (¯ xi (t − τi )) − d´ij gi (xi (t − τi ))} − kjy eyj (t) − η y sgn(eyj (t)),     i=1       y y − y y −  ∆ej (tk ) = eyj (t+ k ) − ej (tk ) = µ ej (tk ), j = 1, 2, · · ·, m; k = 1, 2, · · ·

(9)

where µx , µy are impulsive effect which is given later. The initial conditions of system (9) is ex (s) = ϕ1 (s) − φ1 (s) and ey (s) = ϕ2 (s) − φ2 (s). Assumption1: Assume that there exist positive constants Fj , Gi satisfying |fj (y) − fj (x)| 6 Fj |y − x|

∀x, y ∈ R

|gi (y) − gi (x)| 6 Gi |y − x|

∀x, y ∈ R

8

In Assumption 1, the functions fj , gi need to satisfy the Lipschitz condition, which is a sufficient and unnecessary condition. In addition, the most appropriate Lipschitz constants Fj , Gi need to be selected by other methods. Assumption1 has some limitations. Lemma 1. [34] Suppose α ∈ (0, 1), q ∈ R, then Dα uq (t) =

Γ(1 + q) uq−α (t)Dα u(t). Γ(1 + q − α)

Lemma 2. [16] If the Caputo fractional derivative Dα (u(t)) is integrable, then Itα0 Dα u(t) = u(t) −

n−1 X k=0

u(k) (t0 ) (t − t0 )k k!

Especially, for 0 < α 6 1, one has Itα0 Dα u(t) = u(t) − u(t0 ). Lemma 3. [25] Assume that a positive definite, continuous function V (t) satisfies the following fractional order inequality Dα V (t) 6 −cV q (t) where c > 0, 0 < q < α are all constants. Then V (t) satisfies the following inequality V α−q 6 V α−q (t0 ) −

cΓ(1 + α − q)(t − t0 )α , t0 6 t 6 t1 Γ(1 + α)Γ(1 − q)

and V (t) = 0 for all t > t1 , here t1 is given by Γ(1 + α)Γ(1 − q)V α−q (t0 )
α1 cΓ(1 + α − q)

t1 = t0 +

Lemma 4. [38] If y1 , y2 , · · ·, yn are real numbers, 0 < p 6 1, q > 1, then the following inequalities hold n X i=1

n X i=1

n X |yi |p > ( |yi |)p i=1

n X |yi |q > n(1−q) ( |yi |)q i=1

Lemma 5. Under the Assumption1, then the following inequalities were established: sgn(exi (t))(−` µi x ¯i (t) + µ ´i xi (t)) 6 −µi |exi (t)| + Ti |µ?i − µ?? i | sgn(exi (t))(−` νi y¯j (t) + ν´j yj (t)) 6 −ν j |eyj (t)| + Tej |νj? − νj?? |

9

Proof. 1) For |xi (t)| < Ti and |¯ xi (t)| < Ti , we have sgn(exi (t))(−` µi x ¯i (t) + µ ´i xi (t)) = −sgn(exi (t))(µ?i x ¯i (t) − µ?i xi (t)) = −µ?i |exi (t)| 6 −µi |exi (t)| + Ti |µ?i − µ?? i | 2) For |xi (t)| > Ti and |¯ xi (t)| > Ti , we have sgn(exi (t))(−` µi x ¯i (t) + µ ´i xi (t)) = −sgn(exi (t))(µ?? ¯i (t) − µ?? i x i xi (t)) x ? ?? x = −µ?? i |ei (t)| 6 −µi |ei (t)| + Ti |µi − µi |

3) For |xi (t)| 6 Ti and |¯ xi (t)| ≥ Ti , we have sgn(exi (t))(−` µi x ¯i (t) + µ ´i xi (t)) = −sgn(exi (t))[` µi exi (t) + (` µi xi (t) − µ ´i xi (t))] 6 −µi |exi (t)| + Ti |µ?i − µ?? i | 4) For |xi (t)| ≥ Ti and |¯ xi (t)| 6 Ti , we have sgn(exi (t))(−` µi x ¯i (t) + µ ´i xi (t)) = −sgn(exi (t))[(` µi x ¯i (t) − µ ´i x ¯i (t)) + µ ´i exi (t)] 6 −µi |exi (t)| + Ti |µ?i − µ?? i | Similarly, we have

The proof is completed.

sgn(exi (t))(−` νi y¯j (t) + ν´j yj (t)) 6 −ν j |eyj (t)| + Tej |νj? − νj?? |

Lemma 6. Under the Assumption1, then the following inequalities were established: |` aji fj (¯ yj (t)) − a ´ji fj (yj (t))| 6 aji Fj |eyj (t)| + |a?ji − a?? ji |(Fj Tj + |fj (0)|), |`bji fj (¯ yj (t − τj )) − ´bji fj (yj (t − τj ))| 6 bji Fj |eyj (t − τj )| + |b?ji − b?? ji |(Fj Tj + |fj (0)|), e |` cij gi (¯ xi (t)) − c´ij gi (xi (t))| 6 cij Gi |exi (t)| + |c?ij − c?? ij |(Gi Ti + |gi (0)|),

e |d`ij gi (¯ xi (t − τi )) − d´ij gi (xi (t − τi ))| 6 dij Gi |exi (t − τi )| + |d?ij − d?? ij |(Gi Ti + |gi (0)|).

The proof of these inequalities is similar to the proof of Lemma 1, which is omitted here.

Remark 1. In the existing literature on fractional order memristive neural networks[35, 36, 37], many restrictions are added to the driving function, such as fj (Tj ) = gj (Tej ) = 0 or f (0) = g(0) = 0 (Tj , Tej are the

memristive switching jump). These constraints of the driving function are not required in this paper.

10

4

Main results

This section, some sufficient conditions are obtained to ensure finite time impulsive synchronization of the systems (1) and (2). Theorem 1. Under Assumption 1, for the impulsive sequence {tk }, if there exist impulsive control gain kix , kjy , and positive constants η x , η y such that the following inequalities are established η x − {Ti |µ?i − µ?? i |+ +

m X j=1

+

i=1

j=1

|a?ji − a?? ji |(Fj Tj + |fj (0)|)

bji Fj (|ϕ2j (s) − φ2j (s)|) +

η y − {Tej |νj? − νj?? | + n X

m X

n X i=1

m X j=1

j=1

(10) |b?ji

−

b?? ji |(Fj Tj

e |c?ij − c?? ij |(Gi Ti + |gi (0)|)

dij Gi (|ϕ1i (s) − φ1i (s)|) +

µi + kix −

m X

n X i=1

|d?ij

−

(cij + dij )Gi > 0, ν j + kjy −

e d?? ij |(Gi Tj

+ |fj (0)|)} > 0,

(11) + |gj (0)|)} > 0,

n X (aji + bji )Fj > 0

(12)

i=1

0

0<µ e(α−α ) < 1

then the fractional order MBAMNNs (1) with the impulsive controller (8) is synchronized onto system (2) in a finite time t1 estimated by t1 = t0 + where µ e = max{1+µx , 1+µy }, ξ ∗ =


1 Γ(1 − α0 )Γ(1 + α) (α−α0 ) V (e(t0 )) α ∗ 0 ξ Γ(1 + α − α ) {ξix , ξjy }, V (t0 ) =

min

16i6n,16j6m

are the initial states.

Pn

i=1

|exi (t0 )|+

Pm

j=1

|eyj (t0 )|, exi (t0 ), eyj (t0 )

Proof. Consider Lyapunov function candidate as follows

V (t) = V1 (t) + V2 (t),

(13)

where V1 (t) =

n X i=1

|exi (t)|, V2 (t) =

m X j=1

|eyj (t)|,

For t ∈ (tk−1 , tk ], the Caputo fractional order derivative of V1 (t) can be calculated as Dα V1 (t) 6

n X

sgn(exi (t))Da exi (t),

i=1

11

Based on the Assumptions 1, Lemma 5 and Lemma 6, we have n X

Dα V1 (t) 6

sgn(exi (t)){−` µi x ¯i (t) + µ ´i xi (t) +

i=1

+

m X j=1

[` aji fj (¯ yj (t)) − a ´ji fj (yj (t))]

m X 0 [`bji fj (¯ yj (t − τj )) − ´bji fj (yj (t − τj ))] − kix exi (t) − sgn(exi (t))(ξix |exi (t)|α + η x )} j=1

j=1

i=1

+

(14)

m n X X | + [aji Fj |eyj (t)| + |a?ji − a?? {−µi |exi (t)| + Ti |µ?i − µ?? i ji |(Fj Tj + |fj (0)|]

6

m X j=1

0

x x x x α [bji Fj |eyj (t − τj )| + |b?ji − b?? − η x }, ji |(Fj Tj + |fj (0)|] − ki |ei (t)| − ξi |ei (t)|

On the other hand eyj (t − τj ) 6

sup |eyj (t)| 6

−τ 6s6t

sup |eyj (t)| + sup |eyj (t)| = |ϕ2j (s) − φ2j (s)| + |eyj (t)|,

−τ 6s60

(15)

06s6t

It follows from (14)(15) and condition (10) that Dα V1 (t) 6

m n X X | + [aji Fj |eyj (t)| + |a?ji − a?? {−µi |exi (t)| + Ti |µ?i − µ?? i ji |(Fj Tj + |fj (0)|)] i=1 m X

+

j=1

=

n X

+

j=1

6

i=1

0

x x x x α [bji Fj (|ϕ2j (s) − φ2j (s)| + |eyj (t)|) + |b?ji − b?? − ηx } ji |(Fj Tj + |fj (0)|] − ki |ei (t)| − ξi |ei (t)|

{(−µi − kix )|exi (t)| +

i=1 m X

n X

j=1

m m X X |a?ji − a?? (aji + bji )Fj |eyj (t)| + [Ti |µ?i − µ?? | + ji |(Fj Tj + |fj (0)|) i j=1

j=1

bji Fj (|ϕ2j (s) − φ2j (s)|) +

m X j=1 0

0

x x x α |b?ji − b?? ji |(Fj Tj + |fj (0)|) − η ] − ξi |ei (t)| }

{(−µi − kix )|exi (t)| − ξix |exi (t)|α +

m X (aji + bji )Fj |eyj (t)|},

(16)

j=1

Similarly, for t ∈ (tk−1 , tk ], the fractional order derivative of V2 (t) can be obtained as: Dα V2 (t) 6

m X j=1

0

{(−ν j − kjy )|eyj (t)| − ξjy |eyj (t)|α +

n X i=1

(cij + dij )Gi |exi (t)|},

(17)

Then, for t ∈ (tk−1 , tk ], the derivative of V (t) along trajectories of error system (9) yields Dα V (t) 6

n m X X 0 {(−µi − kix )|exi (t)| − ξix |exi (t)|α + (aji + bji )Fj |eyj (t)|} i=1 m X

+

j=1

6

0

j=1 n X

{(−ν j − kjy )|eyj (t)| − ξjy |eyj (t)|α +

i=1

(cij + dij )Gi |exi (t)|}

n m m n X X X X (−µi − kix + (cij + dij )Gi )|exi (t)| + {(−ν j − kjy + (aji + bji )Fj )|eyj (t)| i=1

−

n X i=1

j=1

0

ξix |exi (t)|α −

m X j=1

j=1

0

ξjy |eyj (t)|α ,

12

i=1

(18)

Let µi + kix −

Let ξ ∗ =

Pm

j=1 (cij

+ dij )Gi > 0, ν j + kjy − Dα V (t) 6 −

Pn

n X i=1

i=1 (aji

+ bji )Fj > 0, from (18), we can obtain 0

ξix |exi (t)|α −

m X j=1

0

ξjy |eyj (t)|α ,

(19)

{ξix , ξjy }, from Lemma 3, we have

min

16i6n,16j6m

n m X X 0 Dα V (t) 6 −ξ ∗ ( |exi (t)| + |eyj (t)|)α i=1

j=1

(20)

0

= −ξ ∗ V α (t), For inequality (20), there must be a nonnegative function h(t) such that 0

0

Dα V (t, e(t)) + h(t)V α (t, e(t)) = −ξ ∗ V α (t, e(t)), multiplying

Γ(1+α−α0 ) −α0 (t, e(t)) Γ(1−α0 ) V

(21)

on both sides of (21), one has

Γ(1 + α − α0 ) −α0 Γ(1 + α − α0 ) Γ(1 + α − α0 ) V (t, e(t))Dα V (t, e(t)) + h(t) = −ξ ∗ , 0 0 Γ(1 − α ) Γ(1 − α ) Γ(1 − α0 ) It can be deduced from Lemma 1 that 0

D(α−α ) V (t, e(t)) = −

h(t)Γ(1 + α − α0 ) ξ ∗ Γ(1 + α − α0 ) − , Γ(1 − α0 ) Γ(1 − α0 )

Based on Lemma 2, integrals from tk−1 to t of the upper formula can be obtained. 0

0

V (α−α ) (t, e(t)) − V (α−α ) (t, e(tk−1 )) = −Itα

∗ 0 h(t)Γ(1 + α − α0 ) α ξ Γ(1 + α − α ) − I , t Γ(1 − α0 ) Γ(1 − α0 )

(22)

From the definition 1 of fractional integra, we can know that Itα

h(t)Γ(1 + α − α0 ) Γ(1 + α − α0 ) = 0 Γ(1 − α ) Γ(1 − α0 )Γ(α)

Z

t

tk−1

h(t) dτ, (t − τ )1−α

(23)

where (t − τ )1−α > 0, Γ(s) > 0, ∀s > 0 and h(τ ) > 0 for ∀τ ∈ (tk−1 , t], we are easy to obtain Itα

h(t)Γ(1 + α − α0 ) > 0. Γ(1 − α0 )

In addition, Itα

ξ ∗ Γ(1 + α − α0 ) ξ ∗ Γ(1 + α − α0 ) = 0 Γ(1 − α ) Γ(1 − α0 )Γ(α)

Z

t

tk−1

1 dτ (t − τ )1−α

ξ ∗ Γ(1 + α − α0 )(t − tk−1 )α ξ ∗ Γ(1 + α − α0 )(t − tk−1 )α = = Γ(1 − α0 )Γ(α)α Γ(1 − α0 )Γ(1 + α)

Substituting (23), (24) into (22), one has 0

0

V (α−α ) (t, e(t)) 6 V (α−α ) (t, e(tk−1 )) −

ξ ∗ Γ(1 + α − α0 )(t − tk−1 )α , t ∈ (tk−1 , tk ] Γ(1 − α0 )Γ(1 + α)

13

(24)

f= Let M

ξ ∗ Γ(1+α−α0 ) Γ(1−α0 )Γ(1+α) ,

0

0

f(t − tk−1 )α , t ∈ (tk−1 , tk ] V (α−α ) (t, e(t)) 6 V (α−α ) (t, e(tk−1 )) − M

(25)

When t = tk , from (9) and (13), one has

V (e(t+ k )) =

n m X X (1 + µx )|exi (t)| + (1 + µy )|eyj (t)| i=1

Let µ e = max{1 + µx , 1 + µy }, one obtains

j=1

V (e(t+ eV (tk ), k )) 6 µ

(26)

From (26) and 0 < α − α0 < 1, we can easily get 0

For t ∈ (t0 , t1 ], from (25), one has

0

0

V (α−α ) (e(t+ e(α−α ) V (α−α ) (tk ), k )) 6 µ 0

0

α f V (α−α ) (t, e(t)) 6 V (α−α ) (t, e(t+ 0 )) − M (t − t0 ) ,

which leads to

Also from (26), one has

0 0 α f V (α−α ) (e(t1 )) 6 V (α−α ) (e(t+ 0 )) − M (t1 − t0 ) ,

 0 0  0 α f V (α−α ) (e(t+ e(α−α ) V (α−α ) (e(t+ 1 )) 6 µ 0 )) − M (t1 − t0 ) ,

For t ∈ (t1 , t2 ], in the same way, we have

0 0 α f V (α−α ) (e(t)) 6 V (α−α ) (e(t+ 1 )) − M (t − t1 )

 0  0 α f f(t − t1 )α 6µ e(α−α ) V (α−α ) (e(t+ −M 0 )) − M (t1 − t0 )

0 α f 6 V (α−α ) (e(t+ 0 )) − M (t − t0 )

 0  0 α α f f 6µ e(α−α ) V (α−α ) (e(t+ 0 )) − M (t1 − t0 ) − M (t − t1 )

0

0

 0  0 α f =µ e(α−α ) V (α−α ) (e(t+ 0 )) − M (t − t0 ) , 0

Since µ e(α−α ) V (α−α ) (e(t)) 6 V (α−α ) (e(t)), one can obtains

0 0 α f V (α−α ) (e(t)) 6 V (α−α ) (e(t+ 0 )) − M (t − t0 )

14

(27)

In general, for any t ∈ (tk , tk+1 ], we find that 0

0

0

0

f(t1 − t0 )α 6 µ ek(α−α ) V (α−α ) (e(t+ ek(α−α ) M 0 )) − µ

V (α−α ) (e(t))

0

0

f(t2 − t1 )α − . . . − µ f(tk−1 − tk−2 )α −e µ(k−1)(α−α ) M e2(α−α ) M 0

f(tk − tk−1 )α − M f(t − tk )α −e µ(α−α ) M 0

0

0

f(t1 − t0 )α 6 µ ek(α−α ) V (α−α ) (e(t+ ek(α−α ) M 0 )) − µ 0

0

f(t2 − t1 )α − . . . − µ f(tk−1 − tk−2 )α −e µk(α−α ) M ek(α−α ) M 0

0

f(tk − tk−1 )α − µ f(t − tk )α −e µk(α−α ) M ek(α−α ) M

 0  0 α f 6 µ ek(α−α ) V (α−α ) (e(t+ 0 )) − M (t − t0 )

Then, for any t ∈ (tk , tk+1 ], we have 0

V (α−α ) (e(t))

0 f(t − t0 )α 6 V (α−α ) (e(t0 )) − M 0

= V (α−α ) (e(t0 )) −

ξ ∗ Γ(1 + α − α0 ) (t − t0 )α . Γ(1 − α0 )Γ(1 + α)

it follows from Lemma 3 that the response system (2) and the drive system (1) can realize finite time synchronization under the impulsive control law (8), and the setting time t1 = t0 +

 Γ(1 − α0 )Γ(1 + α) (α−α0 ) 1 V (e(t0 )) α ∗ 0 ξ Γ(1 + α − α )

This completes the proof. Remark 2. We design impulsive control method to ensure the finite time synchronization of fractional order MBAMNNs with switching jumps mismatch. Recent years, there are some results concerning finite time synchronization or stability for fractional order network systems[26, 30, 29]. However, these papers did not give a setting time value. In [25], by using the properties of fractional order differential and fractional order comparison principle, the criterion of finite time stability for fractional order systems is established, but the finite time synchronization under impulse control is not considered. Obviously, this paper, our results supplement and extend these results in [25, 26, 30, 29]. Remark 3. It can be concluded from the proof of the Theorem 1 that the first term kix , kjy of the controller in Eq.(8) guarantee the synchronization of fractional order MBAMNNs, and show that the synchronization time is infinite. The second term ξix , ξjy are based on the finite time control strategy, which can synchronize the

15

fractional order MBAMNNs in a finite time. The term η x , η y are designed to weaken conditions of the driver function. Corollary 1. Based on Theorem 1, the finite time impulsive synchronization standard of fractional order system can be obtained. If the comparison system of fractional order error system (9) satisfies     v α (t) 6 −pv q (t), t 6= tk ,      v(t+ k ) = −βv(tk ),         v(t0 ) > β0

(28)

where p > 0, 0 < q < α, 0 < β < 1, the error (9) is the finite time impulsive synchronization. When α = 1, the above results are suitable. In particular, if we assume that the driving function satisfies constraint fj (Tj ) = gj (Tej ) = 0 or f (0) =

g(0) = 0, the following Corollary 2 can be obtained according to Theorem 1 and Lemma 3.2 in [37].

Corollary 2. Under Assumption 1, for the impulsive sequence {tk }, if there exist impulsive control gain kix , kjy such that the following inequalities are established µi + kix −

Pm

ν j + kjy −

j=1 (cij

Pn

+ dij )Gi > 0,

(29)

− bji )Fj > 0

(30)

i=1 (aji 0

0<µ e(α−α ) < 1

then the fractional order MBAMNNs (1) with the impulsive controller (8) is synchronized onto system (2) in a finite time t1 estimated by t1 = t0 + where µ e = max{1+µx , 1+µy }, ξ ∗ =


1 Γ(1 − α0 )Γ(1 + α) (α−α0 ) V (e(t0 )) α ξ ∗ Γ(1 + α − α0 ) min

{ξix , ξjy }, V (t0 ) =

16i6n,16j6m

are the initial states.

Pn

i=1

|exi (t0 )|+

Pm

j=1

|eyj (t0 )|, exi (t0 ), eyj (t0 )

Remark 4. Based on Theorem 1 and Corollary 2, it can be seen that the convergence time of the system (9) is affected by order α, α0 and ξ ∗ , the bigger ξ ∗ is, the faster it would be to achieve finite time impulsive synchronization of system (9).

16

5

Numerical simulations

This section, we give numerical simulations examples to verify the effectiveness of Theorem 1. Consider the following fractional order MBAMNNs.  2 2 X X   α  D x (t) = −µ (x (t))x (t) + a (x (t))f (y (t)) + bji (xi (t))fj (yj (t − τj )) + Ii ,  i i i i ji i j j   j=1

j=1

2 2  X X   α  D y (t) = −ν (y (t))y (t) + c (y (t))g (x (t)) + dij (yj (t))gi (xi (t − τi )) + Jj ,  j j j j ij j i i  i=1

(31)

i=1

where α = 0.95, i = 1, 2, j = 1, 2, T1 = T2 = 1, Te1 = Te2 = 2,

µ?1

? ?? ? = 1, µ?? 1 = 1.5, µ2 = 0.8, µ2 = 1, a11 =

? ?? ? ?? ? ?? ? ?? 1.4, a?? 11 = 1.4, a12 = −0.4, a12 = −0.22, a21 = 0.8, a21 = 0.8, a22 = −0.9, a22 = 1.1, b11 = −1.4, b11 = −1.2, ? ?? ? ?? ? ?? ? ?? ? b?12 = 0.2, b?? 12 = −0.1, b21 = 0.5, b21 = −0.2, b22 = 0.7, b22 = 0.7, ν1 = 0.8, ν1 = 1, ν2 = 0.9, ν2 = 1.1, c11 = ? ?? ? ?? ? ?? ? ?? 1.81, c?? 11 = 2.2, c12 = −0.14, c12 = 0.12, c21 = −0.9, c21 = −2.2, c22 = 0.1, c22 = 0.1, d11 = −0.95, d11 = −1.3, ? ?? ? ?? d?12 = 0.08, d?? 12 = 0.15, d21 = −0.2, d21 = −0.18, d22 = −0.3, d22 = −0.3.

Ii = Jj = 0, τ = 1, f1 (σ) = f2 (σ) = g1 (σ) = g2 (σ) =

|σ+1|−|σ−1| , 2

Fj = Gi = 1, the average impulsive

interval is less than T = 0.02. The initial conditions of fractional order MBAMNNs are φ1 (s) = (2, 1)T , φ2 (s) = (−1, 0.2)T , s ∈ [−1, 0]. Fig.2, Fig.3 describes the chaotic attractors of fractional order MBAMNNs (31). The response system is given by :  2 2 X X   α  D x ¯ (t) = −µ (¯ x (t))¯ x (t) + a (¯ x (t))f (¯ y (t)) + bji (¯ xi (t))fj (¯ yj (t − τj )) + Ii + ui (t),  i i i i ji i j j   j=1

j=1

2 2  X X   α  D y ¯ (t) = −ν (¯ y (t))¯ y (t) + c (¯ y (t))g (¯ x (t)) + dij (¯ yj (t))gi (¯ xi (t − τi )) + Jj + vj (t),  j j j j ij j i i  i=1

(32)

i=1

where ui (t) and vj (t)are controllers. The initial conditions of fractional order MBAMNNs (32) are ϕ1 (s) = (1, 2)T , ϕ2 (s) = (1, −1)T , s ∈ [−1, 0]. Set η x > 5.831, η y > 5.881, k1x > 2.79, k2x > 2, k1y > 3.3, k2y > 1.6 and select the impulsive effect µx = −0.15, µy = −0.85, α0 = 0.65, by simple computing, we can get inequalities 0

(10), (11), (13) and 0 < µ e(α−α ) < 1, which are all established. From Theorem 1, we obtain that the response

systems (32) under control (8) is finite timely synchronized with the systems (31), and the estimated setting time is derived as t1


1 Γ(1 − α0 )Γ(1 + α) (α−α0 ) V (e(t0 )) α ∗ 0 ξ Γ(1 + α − α )
1 Γ(1 − 0.65)Γ(1 + 0.95) (0.95−0.65) 0+ V (e(t0 )) 0.95 ξ ∗ Γ(1 + 0.95 − 0.65)
1 Γ(0.35)Γ(1.95) 3.744 0.95 ∗ ξ Γ(1.3)

= t0 + = =

17

(33)

From (33), by choosing different ξ ∗ value, we can obtain the convergence time t1 of the system. The larger ξ ∗ is, the faster the convergence rate of the system is. Fig.4-Fig.5 present time response curves of error between fractional order MBAMNNs (31) and (32). Fig.6-Fig.8 depict synchronization regain with respect to fractional order α and controller parameters. If other parameters are fixed, from Fig.6, we have that with the increasing ξ ∗ , the time t1 is becoming smaller. If other parameters are fixed, from Fig.7 and Fig.8, we know that the time t1 is becoming bigger with the increasing of fractional order α. 2.5

2

x2

1.5

1

0.5

0 -1.5

-1

-0.5

0

0.5

1

1.5

2

x1

Figure 2: The chaotic behavior of fractional order MBAMNNs (31).

2.5

2

y2

1.5

1

0.5

0 -1.5

-1

-0.5

0

0.5

1

1.5

2

y1

Figure 3: The chaotic behavior of fractional order MBAMNNs (31).

6

Conclusion

In this paper, the finite time synchronization between two fractional order MBAMNNs with time delay is discussed. A impulsive control strategy for fractional order MBAMNNs is presented, and a finite time impulsive stability results of fractional order systems is proposed. Based on the fractional comparison principle and the

18

1 ex1 (t) ex2 (t)

e xi(t),i=1,2

0.5

0

-0.5

-1

-1.5 -10

0

10

20

30

40

50

t

Figure 4: Time responses of the errors between systems MBAMNNs (31) and MBAMNNs (32).

2 ey1 (t)

1.5

ey2 (t)

1

e yi(t),i=1,2

0.5 0 -0.5 -1 -1.5 -2 -2.5 -3

0

5

10

15

20

25

30

35

40

45

50

55

t

Figure 5: Time responses of the errors between systems MBAMNNs (31) and MBAMNNs (32).

4 3.5

Setting time t 1

3 2.5 2

Synchronization region

1.5 1 0.5 0 10 5 *

00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 6: The synchronization region with response to fractional order α and parameter ξ ∗ .

19

10

9

8

7

Setting time t 1

6

5

Synchronization region

4

3

2

1

0 0

0.2

0.4

0.6

0.2

0.4

0.6

0.8

0.8

'

0

Figure 7: The synchronization region with response to fractional order α0 and α.

0.6

0.5

Setting time t 1

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 8: The synchronization region with response to fractional order α.

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properties of fractional calculus, sufficient conditions for achieving finite time synchronization of fractional order MBAMNNs are obtained by using some analysis techniques. By analyzing the expression form of setting time, the factors affecting the convergence speed of synchronization error system are discussed. Finally, numerical simulation example is presented to demonstrate the effectiveness of the theoretical results. In future, how to design intermittent control and sampling control to discuss finite time synchronization criteria for fractional order MBAMNNs is our next research direction.

References [1] Huang X, Lin W, Yang B. Global finite-time stabilization of a class of uncertain nonlinear systems, Automatica 41(5) (2005) 881-888. [2] Lu W, Liu X, Chen T. A note on finite-time and fixed-time stability, Neural Networks 81 (2016) 11-15. [3] Ni J, Liu L, Liu C, et al. Fast fixed-time nonsingular terminal sliding mode control and its application to chaos suppression in power system, IEEE Transactions on Circuits and Systems II: Express Briefs 64(2) (2017) 151-155. [4] Zhang D, Cheng J, Cao J, et al. Finite-time synchronization control for semi-Markov jump neural networks with mode-dependent stochastic parametric uncertainties, Applied Mathematics and Computation 344 (2019) 230-242. [5] Kosko B. Bidirectional associative memories, IEEE Transactions on Systems, man, and Cybernetics 18(1) (1988) 49-60. [6] Liu X, Jiang N, Cao J, et al. Finite-time stochastic stabilization for BAM neural networks with uncertainties, Journal of the Franklin Institute 350(8) (2013) 2109-2123. [7] Abdurahman A, Jiang H, Teng Z. Finite-time synchronization for memristor-based neural networks with time-varying delays, Neural Networks 69 (2015) 20-28. [8] Chua L O, Kang S M. Memristive devices and systems, Proceedings of the IEEE 64(2) (1976) 209-223. [9] Strukov D B, Snider G S, Stewart D R, et al. The missing memristor found, nature 453(7191) (2008) 80. [10] Chua L. Resistance switching memories are memristors, Applied Physics A 102(4) (2011) 765-783.

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[11] Tour J M, He T. The fourth element, Nature 453(7191)(2008) 42-43. [12] Hu J, Wang J. Global uniform asymptotic stability of memristor-based recurrent neural networks with time delays, Neural Networks The 2010 International Joint Conference on IEEE (2010) 1-8. [13] Sharifi M J, Banadaki Y M. General SPICE models for memristor and application to circuit simulation of memristor-based synapses and memory cells, Journal of Circuits, Systems, and Computers 19(02) (2010) 407-424. [14] Huang Y, Zhang H, Wang Z. Multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions, Applied Mathematics and Computation 219(3) (2012) 899-910. [15] Mathiyalagan K, Park J H, Sakthivel R. Synchronization for delayed memristive BAM neural networks using impulsive control with random nonlinearities, Applied Mathematics and Computation 259 (2015) 967-979. [16] Pldlubny I. Fractional Differential Equations, Academic press New York (1999). [17] Bagley R L. Torvik P J, On the fractional calculus model of viscoelastic behavior, Journal of Rheology 30(1) (1986) 133-155.

[18] Mohammadzadeh A, Ghaemi S, Kaynak O, Robust predictive synchronization of uncertain fractional-order time-delayed chaotic systems, Soft Computing 23(16) (2019) 6883-6898.

[19] Mohammadzadeh A, Ghaemi S, Optimal synchronization of fractional-order chaotic systems subject to unknown fractional order, input nonlinearities and uncertain dynamic using type-2 fuzzy CMAC, Nonlinear Dynamics 88(4) (2017) 2993-3002. [20] Chen W, Ye L, Sun H. Fractional diffusion equations by the Kansa method, Computers and Mathematics with Applications 59(5) (2010) 1614-1620. [21] Wu A, Zeng Z, Song X. Global Mittag-Leffler stabilization of fractional-order bidirectional associative memory neural networks, Neurocomputing 177 (2016) 489-496.

22

[22] Wang F, Yang Y, Xu X, et al. Global asymptotic stability of impulsive fractional-order BAM neural networks with time delay, Neural Computing and Applications 28(2) (2017) 345-352. [23] Zhang L, Yang Y. Different impulsive effects on synchronization of fractional-order memristive BAM neural networks, Nonlinear Dynamics 93(2) (2018) 233-250. [24] Shen J, Lam J. Non-existence of finite-time stable equilibria in fractional-order nonlinear systems, Automatica 50(2) (2014) 547-551. [25] Li HL, Cao JD, Jiang HJ, et al. Graph theory-based finite-time synchronization of fractional-order complex dynamical networks, Journal of the Franklin Institute 355(13) (2018) 5771-5789. [26] Xiao J, Zhong S, Li Y, et al. Finite-time Mittag-Leffler synchronization of fractional-order memristive BAM neural networks with time delays, Neurocomputing 219 (2017) 431-439. [27] Zhang X, Zhang X, Li D, et al. Adaptive Synchronization for a Class of Fractional Order Time-delay Uncertain Chaotic Systems via Fuzzy Fractional Order Neural Network, International Journal of Control, Automation and Systems 17(5) (2019) 1209-1220. [28] Li HL, Cao J, Jiang H, et al. Finite-time synchronization of fractional-order complex networks via hybrid feedback control, Neurocomputing 320 (2018) 69-75. [29] Velmurugan G, Rakkiyappan R, Cao J. Finite-time synchronization of fractional-order memristor-based neural networks with time delays, Neural Networks 73 (2016) 36-46. [30] Zheng M, Li L, Peng H, et al. Finite-time projective synchronization of memristor-based delay fractionalorder neural networks, Nonlinear Dynamics89(4) (2017) 2641-2655. [31] Li D, Zhang X, Impulsive synchronization of fractional order chaotic systems with time-delay, Neurocomputing 216 (2016) 39-44. [32] Henderson J, Ouahab A, Fractional functional differential inclusions with finite delay, Nonlinear Analysis, Theory, Methods Applications 70(5) (2009) 2091-2105. [33] Kilbas A A, Marzan S A, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations 41(1) (2005) 84-89.

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[34] Peng X, Wu H, Song K, et al. Global synchronization in finite time for fractional-order neural networks with discontinuous activations and time delays, Neural Networks 94 (2017) 46-54. [35] Bao H, Park J H, Cao J. Adaptive synchronization of fractional-order memristor-based neural networks with time delay, Nonlinear Dynamics 82(3) (2015) 1343-1354. [36] Ding S, Wang Z. Lag quasi-synchronization for memristive neural networks with switching jumps mismatch, Neural Computing and Applications (2016) 1-12. [37] Chen J, Zeng Z, Jiang P. Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks, Neural Networks (2014) 51 1-8. [38] H.K. Khalil, lW Grizzle, Nonlinear Systems (2002).

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Lingzhong Zhang received both the B.S and M.S degrees from Northwest Normal University, Lanzhou, China, in 1999 and 2002, respectively. Now he is working toward the Ph.D. degree at School of Internet of Things, Jiangnan University, Wuxi, China. His current research mainly concerns to dynamic and control of fractional order systems and nonlinear systems. He is currently an associate professor of Changshu Institute of Technology.

Yongqing Yang received the B.S. degree from Anhui Normal University, Wuhu, China, the M.S. degree from Anhui University of Science and Technology, Huainan, China, and the Ph.D. degree f rom Southeast University, Nanjing, China, in 1985, 1992, and 2007, respectively. He is currently a professor of Jiangnan University. He is the author or coauthor of more than 50 journal papers. His research interests include nonlinear systems, neural networks and optimization.

Declaration of Interest Statement We declare that we have no associative interest with other people or organizations that can inappropriately influence our work, there is no other personal interest of any service that could be construed as influencing the position presented in, or the review of, the manuscript entitled``Finite time impulsive synchronization of fractional order memristive BAM neural networks".

Yours sincerely, Lingzhong Zhang