Fixed-time synchronization control of memristive MAM neural networks with mixed delays and application in chaotic secure communication

Chaos, Solitons and Fractals 126 (2019) 85–96

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

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Fixed-time synchronization control of memristive MAM neural networks with mixed delays and application in chaotic secure communication Weiping Wang a,b,c,d,∗, Xiao Jia a,b, Xiong Luo a,b, Jürgen Kurths c,d, Manman Yuan a,b,c,d a

School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing 100083, China Beijing Key Laboratory of Knowledge Engineering for Materials Science, Beijing 100083, China Instiute of Physics, Humboldt-University, Berlin 10099, Germany d Potsdam Institute for Climate Impact Research, Potsdam 14473, Germany b c

a r t i c l e

i n f o

Article history: Received 26 November 2018 Revised 10 May 2019 Accepted 31 May 2019 Available online 8 June 2019 Keywords: Memristor Multidirectional associative memory neural networks (MAMNNs) Fixed-time synchronization Time-varying delays Secure communication

a b s t r a c t In this paper, the fixed-time synchronization control problem of memristive multidirectional associative memory neural networks (MMAMNNs) is considered. Based on the nonlinear and chaos characteristics of memristor, a chaotic model is constructed. And then, utilizing the Lyapunov stability theory, two appropriate controllers are constructed and different activation functions are used. This control method ensures that drive system and response system can achieve synchronization within a fixed time. So, compared with previous studies, it has more practical value. In addition, we present a fixed-time synchronization chaotic encryption method, the chaos characteristic of the model is used to encrypt plaintext, and the decryption of ciphertext is realized based on the synchronization control theories. Finally, several numerical simulations are given to demonstrate the validity of the theories and the chaotic secure communication scheme.

1. Introduction Associative memory is a basic function of human brain and has been a hot topic for researchers. In all associative memory models, multidirectional associative memory neural networks (MAMNNs), proposed by M.Hagiwara [1] in 1990, are more conformed to simulate many to many associations of human brain than the other methods. Although the dynamic behaviors of this neural networks have been widely studied [2,3]. But the weight of this model is still immutable, this obviously does not match the variable synaptic weights of the brain. Fortunately, memristor [4–6] have quite similar properties to the synapses in human brain, and the variability of its resistance is often used to simulate human brain. Then, combing with the characters of memristor, researchers constructed MMAMNNs in order to reflect the variable synaptic weight more clearly than the MAMNNs. As far as we know, there are few researches on their dynamic behaviors. Therefore, in this paper, we will discuss the synchronization of this model. Synchronization has been studied [7–11] and applied in many aspects, such as secure communication [12], image encryption

∗

Corresponding author at: XueYuan 30th Road, Haidian District, Beijing, China E-mail address: [email protected] (W. Wang).

https://doi.org/10.1016/j.chaos.2019.05.041 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

© 2019 Elsevier Ltd. All rights reserved.

[13], medicine [14], etc. It means that the dynamic behavior of a coupling system gradually shows the same state. However, many researches are limited to infinite time synchronization, such as projective synchronization [15], adaptive synchronization [16], exponential synchronization [17] and asymptotical synchronization [18]. That means, these kinds of synchronization will hardly satisfy the synchronization criteria unless the time is infinite. Therefore, the method that can achieve the synchronization within a finite time has significant values to some practical applications. Nevertheless, the time of synchronization from the finite-time control methods depends on the initial errors of the coupling system. Some actual applications often need to know the specific time so as to continue the next step of the project, but the initial errors of the system are often unknown. In order to solve this problem, Polyakov [19] puts forward the concept of fixed-time stability, that is, the time required for synchronization between drive-response systems is fixed and computable, it is independent of the initial errors. The fixed-time synchronization has been explored by many researchers [20–22], but MMAMNNs are seldom involved. We will study the fixed-time synchronization control issue based on the proposed MMAMNNs in this paper. Hence it is necessary to make relevant research. Meanwhile, due to the limited transmission

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speed and network jams, delays will inevitably occur and have significant influences on the dynamic behaviors of a chaotic system [23–26]. So it is necessary to consider mixed time-varying delays in our study. Based on the proposed MMAMNNs with mixed delays, we give a chaotic secure communication scheme. In previous articles, asymptotic synchronization [27], exponential synchronization [28] and finite time synchronization [29] have all been studied for secure communications, but it is impossible to get the exact synchronization time. In contrast, the synchronization time required for the control method considered in this paper is a fixed constant. So that the time for decryption is predictable, and it will greatly improve the reliability of the security communication. Due to the consideration of mixed delays, the stability of the transmission is greatly improved. What’s more, the chaos characteristic of the model ensures that the encryption effect will be great. To sum up, comparing with previous studies, this paper focuses on the fixed-time synchronization of MMAMNNs, the results will make the time of synchronization between the drive-response systems more explicit than before. And based on this achievement, a reliable chaotic secure communication scheme is designed. The main contributions of this paper are as follows: 1. We considered fixed-time synchronization control of coupled MMAMNNs with mixed delays. The settling time is only related to the parameters of controllers and it is independent on the initial errors. 2. Some criteria are obtained to guarantee the synchronization of the coupled MMAMNNs with mixed delays. 3. Based on the proposed chaos characteristic of the model and control laws, we give a chaotic secure communication scheme. The remaining chapters of this article are as follows: In Section 2, the MMAMNNs with mixed delays and some preliminaries are introduced. In Section 3, the theorem of fixed-time synchronization is introduced and proved. Moreover, a chaotic secure communication scheme is given. Section 4 gives some numerical simulations to prove the correctness of the theories and the effectiveness of the communication scheme. Section 5 makes a summary of the full text and gives the main conclusions.

tively, and there are two positive constants τ and ρ , |τ vjri (t)| < τ and |ρ (t)| < ρ ; Iri is the external input constant. The model described by Eq. (1) is improved from [30]. There is no connection between the same layer neurons, but neurons of different layers are connected. Benefiting from this special network structure, information can be transmitted bidirectionally among multi-layered neurons, and then associative memory can be realized. Based on the existing research [31,32], Eq. (1) also consider a variety of delays, and then the problem of fixed-time synchronization control with time delays are studied. For convenience, we use avjri (yri (t)), bvjri (yri (t)), cvjri (yri (t)), fvjri (t) and fv jri (t − τv jri (t )) to represent av jri ( fv j (yv j (t )) − yri (t )), bv jri ( fv j (yv j (t − τv jri (t ))) − yri (t )), cv jri ( fv j (yv j (t )) − yri (t )), fv j (yv j (t )) − yri (t ) and fv j (yv j (t − τv jri (t ))) − yri (t ) respectively. So we can rewrite system (1) as follows:

y˙ ri (t ) = Iri − dri yri (t ) +

nv l     [av jri (yri (t )) fv j yv j (t ) ]

v=1,v=r j=1

+

nv l   [bv jri (yri (t )) fv j (yv j (t − τv jri (t ) ) )]

v=1,v=r j=1

+

nv l  

 cv jri (yri (t ))

v=1,v=r j=1

T

t −ρ (t )

 fv j (yv j (s ) )ds .

(2)

Consider the features of memristor, it is convenient to adopt the following definitions:

av jri (yri (t ) ) =

bv jri (yri (t ) ) =

cv jri (yri (t ) ) =

⎧ 1 ⎨ av jri ,

D− ( fv jri (t )) < 0, D− ( fv jri (t )) > 0,

a2v jri ,

⎩

av jri (t ), ⎧ 1 ⎨ bv jri ,

D− ( fv jri (t )) = 0,

−

D− ( fv jri (t − τv jri (t )) < 0,

b2v jri ,

D− ( fv jri (t − τv jri (t )) > 0,

bv jri (t ),

D− ( fv jri (t − τv jri (t )) = 0,

⎩

−

⎧ 1 ⎨ cv jri ,

D− ( fv jri (t )) < 0, D− ( fv jri (t )) > 0,

cv2 jri ,

⎩

cv jri (t − ),

2. Preliminaries and model description



in which

a1

v jri

, a2

v jri

, b1

v jri

D− ( fv jri (t )) = 0,

, b2

1 2 v jri , cv jri and cv jri are constants. Let

In this section, the MMAMNNs with distributed delay and timevarying delays are introduced. Consider the following system:

ξ (s ) = (y11 (s ), y12 (s ), . . . , ylnl (s ))T , s ∈ ([−τ 1 , 0], Rl ), where τ 1 = max{τ , ρ}, represent the initial values of system (2).

y˙ ri (t ) = Iri − dri yri (t )

We treat system (2) as the drive system, the corresponding response system are described by

nv l  

+

[av jri ( fv j (yv j (t )) − yri (t )) fv j (yv j (t ) )]

y˜˙ ri (t ) = Iri − dri y˜ri (t ) +

v=1,v=r j=1 nv l  

+

v=1,v=r j=1

[bv jri ( fv j (yv j (t − τv jri (t ))) − yri (t ))

+

v=1,v=r j=1

+

nv 

v=1,v=r j=1

 ·

T

t −ρ (t )



+

cv jri ( fv j (yv j (t )) − yri (t ))

nv 

[bv jri (y˜ri (t ) ) fv j (y˜v j (t − τv jri (t ) ) )]

nv l  

 cv jri (y˜ri (t ) )

v=1,v=r j=1

+ uri (t ), r = 1, 2, . . . , l,



fv j (yv j (s ) )ds ,

l 

v=1,v=r j=1

· fv j (yv j (t − τv jri (t ) ) )] l 

nv l   [av jri (y˜ri (t ) ) fv j (y˜v j (t ) )]

(1)

in which yri (t) is voltage; nv is supposed to represent the number of neurons in the field v; fri (y) stands for activation function; l is the quantity of fields in system (1); dri , av jri ( fv j (yv j (t )) − yri (t )), bv jri ( fv j (yv j (t − τv jri (t ))) − yri (t )) and cv jri ( fv j (yv j (t )) − yri (t )) are the synapse connection weights; time-varying delays and distributed delay are represented by τ vjri (t) and ρ (t), respec-



T

t −ρ (t )

 fv j (y˜v j (s ) )ds (3)

where uri (t) is appropriate controller. κ (s ) = (y˜11 (s ), y˜12 (s ), . . . , y˜lnl (s ))T ∈ ([−τ 1 , 0], Rl ) are the initial values of the system above. Based on the drive-response systems (2) and (3), the following error system is introduced:

eri (t ) = y˜ri (t ) − yri (t ), r = 1, 2, . . . , l, i = 1, 2, . . . , nr , then we have

(4)

W. Wang, X. Jia and X. Luo et al. / Chaos, Solitons and Fractals 126 (2019) 85–96

e˙ ri (t ) = dri yri(t ) − dri y˜ri (t ) + Fri (t ) + uri (t ),

Proof. First, we prove the following inequality:

r = 1, 2, . . . , l, i = 1, 2, . . . , nr ,

(5)

in which

Fri (t ) =

nv l   [av jri (y˜ri (t )) fv j (y˜v j (t ))

= av jri (y˜ri (t ) ) fv j (y˜v j (t ) ) − av jri (y˜ri (t ) ) fv j (yv j (t ) )

nv 

[bv jri (y˜ri (t ) ) fv j (y˜v j (t − τv jri (t ) ) )

+av jri (y˜ri (t ) ) fv j (yv j (t ) ) − av jri (yri (t ) ) fv j (yv j (t ) )

v=1,v=r j=1

− fv j (y0 )(av jri (y˜ri (t ) ) − av jri (yri (t ) ) )

−bv jri (yri (t ) ) fv j (yv j (t − τv jri (t ) ) )]   T nv l   + cv jri (y˜ri (t ) ) fv j (y˜v j (s ) )ds v=1,v=r j=1

−cv jri (yri (t ) )

t −ρ (t )



T

t −ρ (t )

(8)

av jri (y˜ri (t ) ) fv j (y˜v j (t ) ) − av jri (yri (t ) ) fv j (yv j (t ) )

−av jri (yri (t )) fv j (yv j (t ))] +

   



≤ auv jri Lv j ev j (t ) + Lv j yv j (t ) − y0

a1v jri − a2v jri . av jri (y˜ri (t ) ) fv j y˜v j (t ) − av jri (yri (t ) ) fv j yv j (t )

Based on Assumption 3, we transform the above inequality as follows:

v=1,v=r j=1

l 

87

= av jri (y˜ri (t ) )( fv j (y˜v j (t ) ) − fv j (yv j (t ) ) ) +(av jri (y˜ri (t ) ) − av jri (yri (t ) ) )( fv j (yv j (t ) ) − fv j (y0 ) ).

(9)

Using Assumption 2, we have



fv j (yv j (s ) )ds .

(6)

The initial values of the error system (4) are ψ (s ) = κ (s ) − ξ (s ). To get the main results, an assumption is introduced: Assumption 1. There is a constant Mri > 0 which satisfies |fri ( · )| ≤ Mri , r = 1, 2, . . . , l, i = 1, 2, . . . , nr , i.e., fri ( · ) is bounded. Assumption 2. The activation function fri (t) is Lipschitz continuous on R, i.e., there is a positive constant Lri , | fri (x ) − fri (y )| ≤ Lri |x − y|, x, y ∈ R, x = y, r = 1, 2, . . . , l, i = 1, 2, . . . , nr . Assumption 3. fri has a zero point in its domain, i.e., ∃ y0 , fri (y0 ) = 0. Lemma 1. If the activation function fri satisfies the conditions in   v Assumption 1, then, |Fri (t)| ≤ ri , ri = lv=1,v=r nj=1 2Mv j (auv jri + u u bv jri + cv jri ρ ), where

   



≤ auv jri Lv j ev j (t ) + Lv j yv j (t ) − y0

a1v jri − a2v jri . av jri (y˜ri (t ) ) fv j y˜v j (t ) − av jri (yri (t ) ) fv j yv j (t )

(10)

By the same method, the following inequalities can be proved:

bv jri (y˜ri (t ) ) fv j (y˜v j (t − τv jri (t ) ) ) − bv jri (yri (t ) ) fv j (yv j (t − τv jri (t ) ) ) ≤ buv jri Lv j |ev j (t − τv jri (t ) )| + Lv j |yv j (t − τv jri (t ) ) − y0 ||b1v jri − b2v jri |, (11) T T cv jri (y˜ri (t ) ) t −ρ (t ) fv j (y˜v j (s ) )ds − cv jri (yri (t ) ) t −ρ (t ) fv j (yv j (s ) )ds T T ≤ cvu jri Lv j t −ρ (t ) |ev j (s )|ds + Lv j |cv1 jri − cv2 jri | t −ρ (t ) |yv j (s ) − y0 |ds. (12)

auv jri = max

1 2 

av jri , av jri ,

Combining the inequalities (8), (11) and (12) with the definition of Fri (t), we can get the inequality (7). The proof of Lemma 2 is completed. 

buv jri = max

1 2 

bv jri , bv jri ,

Lemma 3 [21]. If x1 , x2 , ..., xn ≥ 0, 0 < p ≤ 1, q > 1, then we have n n n  p q p 1−q ( n x )q . i=1 xi ≥ ( i=1 xi ) , i=1 xi ≥ n i=1 i

 cv jri = max cv1 jri , cv2 jri .

Definition 1 [19]. If system (2) and (3) satisfy the following conditions, they will be synchronized within a finite time: ∃ t∗ (e(0)) > 0, where e(t ) = (e11 (t ), e12 (t ), . . . , emnm (t ))T , lim e(t ) = 0 and

Proof. This lemma can be identified easily according to the Assumption 1 and the definition of Fri (t). 

e(t) ≡ 0 for ∀t > t∗ (e(0)).

u

Lemma 2. If the activation function fri (yri (t)) satisfies the conditions in Assumption 2 and 3, then we have the following inequality:

Fri (t ) ≤

nv l   [auv jri Lv j |ev j (t )|

v=1,v=r j=1

nv l   [buv jri Lv j |ev j (t − τv jri (t ) )|

1 . b(q−1 )

(i) V (x ) = 0 ⇔ x = 0; (ii) V˙ (e(t ) ) ≤ −aV p (e(t ) ) − bV q (e(t ) ) is established for all solution e(t) of error system (4), where a, b > 0, 0 < p < 1 and q > 1.

v=1,v=r j=1

+Lv j |yv j (t − τv jri (t ) ) − y0 ||b1v jri − b2v jri |]   T nv l   + cvu jri Lv j |ev j (s )|ds v=1,v=r j=1

+Lv j |cv1 jri − cv2 jri |

t −ρ (t )



T

t −ρ (t )

Definition 2 [19]. If system (2) and (3) satisfy the following conditions, they are said to be synchronized in fixed-time: (i) System (2) and (3) satisfie the definition 1; (ii) There is a positive constant Tmax , for any e(0), t∗ (e(0)) is always less than Tmax . Lemma 4 [19]. V (· ) : Rn → R+ ∪ {0} is a continuous radically unbound function. If the following requirements can be satisfied, Then the error system (4) is fixed-time stable. Moreover, Tmax = a(11−q) +

+Lv j |yv j (t ) − y0 ||a1v jri − a2v jri |] +

t →t ∗ (e (0 ))

3. Main results



|yv j (s ) − y0 |ds ,

(7)

where the definitions of auv jri , buv jri and cvu jri are the same as those in Lemma 1 and fri (y0 ) = 0.

In this section, appropriate synchronization nonlinear controllers are given for activation functions which meets different conditions in system (2) and (3), so that the drive-respone system can achieve synchronization in a fixed time. Moreover, a secure communication scheme will be given.

88

W. Wang, X. Jia and X. Luo et al. / Chaos, Solitons and Fractals 126 (2019) 85–96 p+1 p+1 1 min{h3ri }2 2 (Vr (t ) ) 2 l r,i 1−q 1+q q+1 1 − min{h4ri }l 2 nr 2 (Vr (t ) ) 2 , l r,i

3.1. Nonlinear controller

=−

Theorem 1. Suppose Assumption 1 is true. Constructing the following controller:

and then

uri = −h1ri sign(eri (t ) )|eri (t )| − h2ri sign(eri (t ) )

V˙ (t ) =

−h3ri sign(eri (t ) )|eri (t )| p − h4ri sign(eri (t ) )|eri (t )|q , r = 1, 2, . . . , l, i = 1, 2, . . . , nr ,

Tmax =

α β

−dri , h2ri

(13)

1

α (1 −

p+1 2

)

h3ri

+

h4ri

1

1 2l

Vr (t ), Vr (t ) =

r=1

≤−

l p+1 p+1 1 min{h3ri }2 2 (Vr (t ) ) 2 l r,i r=1

l 1−q q+1 q+1 1 − min{h4ri }l 2 nr 2 (Vr (t ) ) 2 . l r,i

α=

nv 

p+1 1 min{h3ri }2 2 , l r,i

based on the Lemma 3, we get

V˙ (t ) ≤ −α (14)

l 

l 

−β l

i=1

l



1− q+1 2

= −αV (t )

Tmax =

i=1

=

−dri y˜ri (t ) + Fri (t ) + uri (t )]

l 

2

−h4ri

nr nr 1 1 p+1 q+1 − h3ri |eri (t )| − h4ri |eri (t )| . l l

+



(16)



−

1 l

nr 

q+1

h4ri |eri (t )|

i=1

nr  1 ≤ − min{h3ri } |eri (t )|2 l r,i i=1

nr  1 − min{h4ri } |eri (t )|2 l r,i i=1

+

+

.

(19)

1 l β ( q+1 − 1) 2

2 . l β (q − 1 )

T

(20)





− h2ri |ev j (t )| − h3ri |ev j (t − τv jri (t ) )|

 |ev j (s )|ds sign(eri (t ) )

t −ρ (t )

nr l  



With Lemma 2 and the conditions about h1ri and h2ri in Theorem 1, we have p+1

)

v=1,v=r j=1

i=1



q+1 2

v=1,v=r j=1

i=1

i=1

Vr (t )

− β lV (t )

nr l  

nr 1 + |eri (t )|(ri − h2ri ) l

i=1

i=1

p+1 2

α (1 − p)

+

h3ri |eri (t )|

p+1 2

1

α (1 −

nr 1 |eri (t )|2 (−h1ri − dri ) l

1 l

 q+1 2

uri (t ) = −h1ri sign(eri (t ) )|eri (t )|

i=1

V˙ r (t ) ≤ −

q+1 2

Theorem 2. Suppose Assumption 2 and 3 are true. Constructing the following controller:

i=1

nr 

(Vr (t ) )

r=1

The proof is completed.

|eri (t )|[−dri |eri (t )| + ri ]

nr 1 + |eri (t )|sign(eri (t ) )uri (t ) l

=

q+1 2

From Lemma 4, this theorem is correct. In addition,

nr 1 |eri (t )|sign(eri (t ) )[dri yri (t ) l

1 l

l

r=1

i=1

≤

l 

Vr (t )

(15) q+1 2

−β

 p+1 2

r=1

(eri (t ) )2 .

p+1 2

(Vr (t ) )

r=1

 ≤ −α

nr 1 V˙ r (t ) = eri (t )e˙ ri (t ) l

nr 

1−q 1 min{min{h4ri }nr 2 }, l r,i r,i

β=

With Lemma 1, the derivative of Vr (t) is as follows

=

(18)

Let

l β ( q+1 − 1) 2

Proof. We construct a Lyapunov function as follows:

V (t ) =

V˙ r (t )

r=1

2 2 = + , α (1 − p) l β (q − 1 )  p+1 1 = min h3ri 2 2 , l r,i   1−q  1 = min min h4ri nr 2 . l r,i r,i

l 

l  r=1

where ≥ ≥ ri , and are positive constants, 0 < p < 1, q > 1. Then the fixed-time synchronization can be achieved for system (2) and (3) under the controller (13). And the synchronization time is within Tmax , where h1ri

(17)

−h7ri

T

t −ρ (t )

−h8ri sign

 − h5ri |yv j (t ) − y0 | − h6ri |yv j (t − τv jri (t ) ) − y0 |

 |yv j (s ) − y0 |ds sign(eri (t ) )

(eri (t ) )|eri (t )| p − h9ri sign(eri (t ) )|eri (t )|q

r = 1, 2, . . . , l, i = 1, 2, . . . , nr ,

(21)

 q+1 2

where

 q+1 2

h9ri are positive constants, 0 < p < 1, q > 1. Then the fixed-time synchronization can be achieved for system (2) and (3) under the controller (21). And the synchronization time is within Tmax , where

h1ri

≥ −dri ,

h2ri

Lv j |a1v jri − a2v jri |, h6ri ≥

1−q

nr 2

Tmax =

1

α (1 −

p+1 2

)

≥ v jri Lv j , h3ri ≥ buv jri Lv j , h4ri ≥ cvu jri Lv j , h5ri ≥ Lv j |b1v jri − b2v jri |, h7ri ≥ Lv j |cv1 jri − cv2 jri |, h8ri and

+

au

1 l β ( q+1 − 1) 2

W. Wang, X. Jia and X. Luo et al. / Chaos, Solitons and Fractals 126 (2019) 85–96

89

where

d11 = 0.35, d21 = 0.66, d31 = 1.66,



0.06, 0.59, a1121 (t − ),

D− ( f1121 (t )) < 0, D− ( f1121 (t )) > 0, D− ( f1121 (t )) = 0,

0.55, −0.78, a1131 (t − ),

D− ( f1131 (t )) < 0, D− ( f1131 (t )) > 0, D− ( f1131 (t )) = 0,

0.69, −0.48, a2111 (t − ),

D− ( f2111 (t )) < 0, D− ( f2111 (t )) > 0, D− ( f2111 (t )) = 0,

0.58, 0.23, a2131 (t − ),

D− ( f2131 (t )) < 0, D− ( f2131 (t )) > 0, D− ( f2131 (t )) = 0,

0.7, −0.1, a3111 (t − ),

D− ( f3111 (t )) < 0, D− ( f3111 (t )) > 0, D− ( f3111 (t )) = 0,

−0.58, −0.85, a3121 (t − ),

D− ( f3121 (t )) < 0, D− ( f3121 (t )) > 0, D− ( f3121 (t )) = 0,

1.43, 0.71, b1121 (t − ),

D− ( f1121 (t − τ1121 (t ))) < 0, D− ( f1121 (t − τ1121 (t ))) > 0, D− ( f1121 (t − τ1121 (t ))) = 0,

0.61, 0.98, b1131 (t − ),

D− ( f1131 (t − τ1131 (t ))) < 0, D− ( f1131 (t − τ1131 (t ))) > 0, D− ( f1131 (t − τ1131 (t ))) = 0,

0.64, 0.55, b2111 (t − ),

D− ( f2111 (t − τ2111 (t ))) < 0, D− ( f2111 (t − τ2111 (t ))) > 0, D− ( f2111 (t − τ2111 (t ))) = 0,

0.65, 0.74, b2131 (t − ),

D− ( f2131 (t − τ2131 (t ))) < 0, D− ( f2131 (t − τ2131 (t ))) > 0, D− ( f2131 (t − τ2131 (t ))) = 0,

0, 0.61, b3111 (t − ),

D− ( f3111 (t − τ3111 (t ))) < 0, D− ( f3111 (t − τ3111 (t ))) > 0, D− ( f3111 (t − τ3111 (t ))) = 0,

0.7, 0.71, b3121 (t − ),

D− ( f3121 (t − τ3121 (t ))) < 0, D− ( f3121 (t − τ3121 (t ))) > 0, D− ( f3121 (t − τ3121 (t ))) = 0,

−0.212, 0.212, c1121 (t − ),

D− ( f1121 (t )) < 0, D− ( f1121 (t )) > 0, D− ( f1121 (t )) = 0,

−0.46, −0.24, c1131 (t − ),

D− ( f1131 (t )) < 0, D− ( f1131 (t )) > 0, D− ( f1131 (t )) = 0,

0.18, −0.112, c2111 (t − ),

D− ( f2111 (t )) < 0, D− ( f2111 (t )) > 0, D− ( f2111 (t )) = 0,

0.06, 0.02, c2131 (t − ),

D− ( f2131 (t )) < 0, D− ( f2131 (t )) > 0, D− ( f2131 (t )) = 0,

0, 0.028, c3111 (t − ),

D− ( f3111 (t )) < 0, D− ( f3111 (t )) > 0, D− ( f3111 (t )) = 0,

a1121 (y21 (t ) ) =



Fig. 1. Secure communication scheme.

a1131 (y31 (t ) ) = =

α= β

2

2

+

α (1 − p) l β (q − 1 ) 8  p+1 1 min hri 2

2



, a2111 (y11 (t ) ) =

,



l r,i   1−q  1 = min min h9ri nr 2 . l r,i r,i

(22)

Proof. Theorem 2 uses the same proof method as Theorem 1 and we omitted it here.  Remark 1. The term sign(eri (t)) in controller (13) and (21) is discontinuity, and it may not be suitable to some practical applicae (t ) tions, so we use continuous term |e ri(t )|+ε to replace it. Moreover, ri

a2131 (y31 (t ) ) =

 a3111 (y11 (t ) ) =

 a3121 (y21 (t ) ) =

the term ε is a small enough positive number.

 b1121 (y21 (t ) ) =

3.2. Chaotic security communication In this subsection, according to the chaotic path of systems (2) and (3), we designed an effective secure communication scheme. Fig. 1 shows the procedure of the scheme. We first introduce plaintext signals pri (t) and adaptive tracking signals δ ri (t) into drive system to generate the corresponding chaotic signals yri (t). Then superimposing yri (t) on pri (t) to construct encrypted transmission signals ωri (t), i.e., ωri (t ) = pri (t ) + ηri (t ). Whereafter, we introduced ωri (t) and adaptive tracking signals δ˜ri (t ) into response system and obtain the corresponding synchronized chaotic signal y˜ri (t ). Then we can get the decrypted plaintex signals ρ ri (t) by subtracting x˜ri (t ) from ωri (t). That is, ρri (t ) = ωri (t ) − x˜ri (t ). The chaotic secret communication is realized by hiding the plaintext in chaotic sequence, and it only takes one neuron to complete the transmission of one signal. This makes the transmission efficient and easy to implement.

 b1131 (y31 (t ) ) =

 b2111 (y11 (t ) ) =

 b2131 (y31 (t ) ) =

 b3111 (y11 (t ) ) =

 b3121 (y21 (t ) ) =



4. Numerical simulation

c1121 (y21 (t ) ) =

In this section, several examples will be given to verify the correctness and effectiveness of the results. Example 1. Theorem 1 will be validated in this example. Consider a three-fields MMAMNNs, and there is one neuron in each field. Which means that l = 3, nv = 1, for v = 1, 2, 3.

 c1131 (y31 (t ) ) =

y˙ r1 (t ) = Ir1 − dr1 yr1 (t ) +

3 

 c2111 (y11 (t ) ) =

av1r1 (yr1 (t ) ) fv1 (yv1 (t ) )

v=1,v=r

+

3 

 bv1r1 (yr1 (t ) ) fv1 (yv1 (t − τv1r1 (t ) ) )

c2131 (y31 (t ) ) =

v=1,v=r

+

3 

v=1,v=r

r = 1, 2, 3,

cv1r1 (yr1 (t ) )



T

t −ρ (t )

 fv1 (yv1 (s ) )ds,

c3111 (y11 (t ) ) = (23)

90

W. Wang, X. Jia and X. Luo et al. / Chaos, Solitons and Fractals 126 (2019) 85–96

Fig. 3. Synchronization error of system (23) and (24) under Assumption 1.

Fig. 2. State trajectories of system (23) and (24) under Assumption 1.

 c3121 (y21 (t ) ) =

0.008, −0.1, c3121 (t − ),

D− ( f3121 (t )) < 0, D− ( f3121 (t )) > 0, D− ( f3121 (t )) = 0.

In this example, we set fr1 (y ) = tanh(y ), r = 1, 2, 3. From Assumption 1, we have Mv1 = 1, v = 1, 2, 3. Let τv jri (t ) = cos(t )/2 + 1/2, ρ (t ) = sin(t )/2 + 1/2. Setting the initial values of the system (23) are ξ (s ) = (ξ11 (s ), ξ21 (s ), ξ31 (s ))T = (2.8, 1.85, 1.37 )T . The response system is as follows:

y˜˙ r1 (t ) = Ir1 − dr1 y˜r1 (t ) + ur1 (t ) +

3 

av1r1 (y˜r1 (t ) ) fv1 (y˜v1 (t ) )

v=1,v=r

+

3 

bv1r1 (y˜r1 (t ) ) fv1 (y˜v1 (t − τv1r1 (t ) ) )

v=1,v=r

+

3 

cv1r1 (y˜r1 (s ) )

v=1,v=r



T

t −ρ (t )

fv1 (y˜v1 (s ) )ds,

(24)

where r = 1, 2, 3, ur1 (t) is controller. Let κ (s ) = (y˜11 (s ), y˜21 (s ), y˜31 (s ))T = (2.5, 1.97, 0.75 )T . The state trajectories without controller are shown in Fig. 2. From Theorem 1, the following inequalities should be established:

h111 ≥ −d11 = −0.35, h211 ≥ 11 = 5.696, h121 ≥ −d21 = −0.66, h221 ≥ 21 = 7.784, h131 ≥ −d31 = −1.66, h231 ≥ 31 = 7.2. h111

h121

(25) h131

h211

h221

So we make = 0.7, = 0.9, = 1.5, = 9, = 12, h231 = 8. Moreover, h3r1 = h4r1 = 1, r = 1, 2, 3, p = .7, q = 3. Then we have the following controllers:

The state trajectories with controller (26) are shown in Fig. 4. Figs. 3 and 5 respectively represent synchronization error of system (23) and (24) without and with controller. The comparison of them show that controller (13) is effective. The error of each neurons decreased gradually within Tmax and finally reached 0, which is coincided with Theorem 1. Through the comparison of Figs. 2 and 4, it can be seen that under the action of the controller, the trajectories of the drive-response system overlap quickly, but there is no trend of overlap in the absence of controller. Example 2. In this example, Theorem 2 will be validated. We use the same system as the one in Example 1, but we will use the activation functions satisfying Assumptions 2 and 3. So we set fr1 (y ) = 0.5sin(y ) + 0.5y, r = 1, 2, 3. From Assumption 2 and 3, we have Lr1 = 1, r = 1, 2, 3, y0 = 0. From Theorem 2, the following inequalities should be established: h111 ≥ −d11 = −0.35, h211 ≥ 0.7,

h311 ≥ 0.64,

h411 ≥ 0.18,

h121 h131 h421 h431

h321 h331

≥ 0.71,

h511 ≥ 1.17,

≥ 0.98,

h611 ≥ 0.61,

≥ −d21 = −0.66, ≥ −d31 = −1.66, ≥ 0.212, ≥ 0.46,

≥ 0.53, ≥ 1.33,

≥ 0.85, ≥ 0.78, h621 h631

h111

≥ 0.72, ≥ 0.37, h121

h711 h731

≥ 0.292, h721 ≥ 0.424, ≥ 0.22

So we make = 0.8, = 1, = −0.5, h211 = 1, h221 = 2 3 3 3 1.5, h31 = 1,h11 = 0.7, h21 = 0.9, h31 = 1.5, h411 = 0.5, h421 = 0.6, h431 = 1,h511 = 2, h521 = 0.8, h531 = 1.7, h611 = 0.9, h621 = 1.2, h631 = 0.4,h711 = 0.55, h721 = 0.6, h731 = 0.3. Moreover, h8r1 = h9r1 = 1.2, r = 1, 2, 3, p = .5, q = 4.

h131

Then we can get the controller satisfying Theorem 2:

(t ) = −0.8sign(e11 (t ) )|e11 (t )| +

3 

[−|ev1 (t )| − 0.7|ev1 (t − τv111 (t ) )|

v=1,v=1

u11 = −0.7sign(e11 (t ) )|e11 (t )| − 9sign(e11 (t ) )



−sign(e11 (t ) )|e11 (t )|0.7 − sign(e11 (t ) )|e11 (t )|3 ,

−0.5

u21 = −0.9sign(e21 (t ) )|e21 (t )| − 12sign(e21 (t ) ) −sign(e21 (t ) )|e21 (t )|0.7 − sign(e21 (t ) )|e21 (t )|3 ,

+

u31 = −1.5sign(e31 (t ) )|e31 (t )| − 8sign(e31 (t ) ) −sign(e31 (t ) )|e31 (t )|0.7 − sign(e31 (t ) )|e31 (t )|3 .

h521 h531

h221 h231

T

t −ρ (t )

3 

|ev1 (s )|ds]sign(e11 (t ) )

[−2|xv1 (t )| − 0.9|xv1 (t − τv111 (t ) )|

v=1,v=1

(26)

On the basis of Theorem 1, system (23) and (24) satisfie the conditions of fixed-time synchronization under the controller (26), moreover, Tmax = 12.0957.



−0.55

T

t −ρ (t )

|xv1 (s )|ds]sign(e11 (t ) )

−1.2sign(e11 (t ) )|e11 (t )|0.5 − 1.2sign(e11 (t ) )|e11 (t )|4 , u21 (t ) = −sign(e21 (t ) )|e21 (t )|

W. Wang, X. Jia and X. Luo et al. / Chaos, Solitons and Fractals 126 (2019) 85–96

Fig. 4. State trajectories of system (23) and (24) with controller (26) under Assumption 1.

Fig. 5. Synchronization error of system (23) and (24) with controller (26) under Assumption 1.

+

3 

[−1.5|ev1 (t )| − 0.9|ev1 (t − τv121 (t ) )|

v=1,v=2



−0.6 +

T

t −ρ (t )

3 

|ev1 (s )|ds]sign(e21 (t ) )

Fig. 6. State trajectories of system (23) and (24) with Assumptions 2–3.

−1.2sign(e21 (t ) )|e21 (t )|0.5 − 1.2sign(e21 (t ) )|e21 (t )|4 , u31 (t ) = 0.5sign(e31 (t ) )|e31 (t )|



[−0.8|xv1 (t )| − 1.2|xv1 (t − τv121 (t ) )|

−0.6

T

t −ρ (t )

|xv1 (s )|ds]sign(e21 (t ) )

[−|ev1 (t )| − 1.5|ev1 (t − τv131 (t ) )|

v=1,v=3

−

v=1,v=2



3 

+

+

T

|ev1 (s )|ds]sign(e31 (t ) )

t −ρ (t ) 3 

[−1.7|xv1 (t )| − 0.4|xv1 (t − τv131 (t ) )|

v=1,v=3

91

92

W. Wang, X. Jia and X. Luo et al. / Chaos, Solitons and Fractals 126 (2019) 85–96

Fig. 7. Synchronization error of system (23) and (24) with Assumptions 2–3.

 −0.3

T

t −ρ (t )

Fig. 9. Synchronization error of system (23) and (24) with controller 27 under Assumptions 2–3.

|xv1 (s )|ds]sign(e31 (t ) )

−1.2sign(e31 (t ) )|e31 (t )|0.5 − 1.2sign(e31 (t ) )|e31 (t )|4 . (27) On the basis of Theorem 2, systems (23) and (24) satisfie the conditions of fixed-time synchronization under the controller (27), moreover, Tmax = 6.5016. The state trajectories of systems (23) and (24) without controller are shown in Fig. 6. We can see that the trajectories of drive system does not coincide with the trajectories of the response system. However, in Fig. 8, under controller (27), we can see that the trajectories of the systems soon coincide with each

other. Figs. 7 and 9 respectively represent synchronization error of system (23) and (24) without and with controller. We can see that under controller (27), the error quickly converges to 0 and the synchronization time is less than Tmax . This shows that the control scheme proposed in this paper is effective.

Example 3. In this example, we will verify the effectiveness of the proposed secure communication scheme in 3.2 subsection. In the same case as Example 1, we use the model containing three fields, and each field has one neuron. Then we can get the following

Fig. 8. State trajectories of system (23) and (24) with controller 27 under Assumptions 2–3.

W. Wang, X. Jia and X. Luo et al. / Chaos, Solitons and Fractals 126 (2019) 85–96

93

where

d11 = 0.87, d21 = 0.34, d31 = 0.55, w1r1 = 0.2, w2r1 =0.5, r = 1, 2, 3, 0.1, D− ( f1121 (t )) < 0, a1121 (y21 (t ) ) = 0.21, D− ( f1121 (t )) > 0, a1121 (t − ), D− ( f1121 (t )) = 0,



0.2, D− ( f1131 (t )) < 0, 0.53, D− ( f1131 (t )) > 0, a1131 (t − ), D− ( f1131 (t )) = 0,

a1131 (y31 (t ) ) =

 a2111 (y11 (t ) ) =

−0.37, D− ( f2111 (t )) < 0, 0.21, D− ( f2111 (t )) > 0, a2111 (t − ), D− ( f2111 (t )) = 0,



−0.92, D− ( f2131 (t )) < 0, 0.2, D− ( f2131 (t )) > 0, a2131 (t − ), D− ( f2131 (t )) = 0,

a2131 (y31 (t ) ) =



0.45, D− ( f3111 (t )) < 0, 0.86, D− ( f3111 (t )) > 0, a3111 (t − ), D− ( f3111 (t )) = 0,

a3111 (y11 (t ) ) =

 a3121 (y21 (t ) ) =

Fig. 10. The chaotic attractors of system (28).

 b1121 (y21 (t ) ) =



0.58, D− ( f3121 (t )) < 0, 0.79, D− ( f3121 (t )) > 0, a3121 (t − ), D− ( f3121 (t )) = 0,

−0.25, D− ( f1121 (t − τ1121 (t ))) < 0, 0.04, D− ( f1121 (t − τ1121 (t ))) > 0, b1121 (t − ), D− ( f1121 (t − τ1121 (t ))) = 0,

−0.15, D− ( f1131 (t − τ1131 (t ))) < 0, −0.1, D− ( f1131 (t − τ1131 (t ))) > 0, b1131 (t − ), D− ( f1131 (t − τ1131 (t ))) = 0,

b1131 (y31 (t ) ) =



0.03, D− ( f2111 (t − τ2111 (t ))) < 0, 0.04, D− ( f2111 (t − τ2111 (t ))) > 0, b2111 (t − ), D− ( f2111 (t − τ2111 (t ))) = 0,

b2111 (y11 (t ) ) =



−0.19, D− ( f2131 (t − τ2131 (t ))) < 0, −0.22, D− ( f2131 (t − τ2131 (t ))) > 0, b2131 (t − ), D− ( f2131 (t − τ2131 (t ))) = 0,

b2131 (y31 (t ) ) =



0.52, D− ( f3111 (t − τ3111 (t ))) < 0, −0.01, D− ( f3111 (t − τ3111 (t ))) > 0, b3111 (t − ), D− ( f3111 (t − τ3111 (t ))) = 0,

b3111 (y11 (t ) ) =



0.54, D− ( f3121 (t − τ3121 (t ))) < 0, 0.67, D− ( f3121 (t − τ3121 (t ))) > 0, b3121 (t − ), D− ( f3121 (t − τ3121 (t ))) = 0,

b3121 (y21 (t ) ) =

 c1121 (y21 (t ) ) = Fig. 11. The plaintext signals pr1 (t), decrypted plaintext signals ρ r1 (t) and encrypted signals ωr1 (t) .

−0.11, D− ( f1121 (t )) < 0, −0.01, D− ( f1121 (t )) > 0, c1121 (t − ), D− ( f1121 (t )) = 0,



0.02, D− ( f1131 (t )) < 0, −0.3, D− ( f1131 (t )) > 0, − −  c1131 (t −), D ( f1131 (t )) = 0, 0.05, D ( f2111 (t )) < 0, c2111 (y11 (t ) ) = −0.14, D− ( f2111 (t )) > 0, c2111 (t − ), D− ( f2111 (t )) = 0, c1131 (y31 (t ) ) =

system for encryption.

⎧ y˙ r1 (t ) = Ir1 − dr1 yr1 (t ) ⎪ ⎪ ⎪ 3  ⎪ ⎪ + av1r1 (yr1 (t ) ) fv1 (yv1 (t ) ) ⎪ ⎪ ⎪ v=1,v=r ⎪ ⎪ 3 ⎪  ⎨ + bv1r1 (yr1 (t ) ) fv1 (yv1 (t − τv1r1 (t ) ) ) v=1,v=r ⎪ 3 ⎪ T  ⎪ ⎪ + cv1r1 (yr1 (t ) ) t −ρ (t ) fv1 (yv1 (s ) )ds ⎪ ⎪ v=1,v=r ⎪ ⎪ ⎪ ⎪ +w1r1 ∗ ( pr1 (t ) − δr1 (t ) ), ⎪ ⎩˙ δr1 (t ) = w2r1 ∗ ( pr1 (t ) − δr1 (t )),



c2131 (y31 (t ) ) =

 (28)

c3111 (y11 (t ) ) =

0.05, D− ( f2131 (t )) < 0, −0.01, D− ( f2131 (t )) > 0, c2131 (t − ), D− ( f2131 (t )) = 0,

0.4, D− ( f3111 (t )) < 0, −0.13, D− ( f3111 (t )) > 0, c3111 (t − ), D− ( f3111 (t )) = 0,

 c3121 (y21 (t ) ) =

0.08, D− ( f3121 (t )) < 0, 0.1, D− ( f3121 (t )) > 0, c3121 (t − ), D− ( f3121 (t )) = 0.

Here we set the initial values of system (28) are ξ (s ) = δr1 (0 ) = 0

(y11 (s ), y21 (s ), y31 (s ))T = (−0.0125, −0.077, 0.0024 )T ,

94

W. Wang, X. Jia and X. Luo et al. / Chaos, Solitons and Fractals 126 (2019) 85–96

Fig. 12. Two examples of decryption failures. (a) Decryption effect without controller. (b) Error between Encryption Sequence and Decryption Sequence without Controller. (c) Encryption effect in the case of controller parameter Error. (d) Error between Encryption Sequence and Decryption Sequence with wrong controller parameters.

Table 1 Sensitivity to small changes in initial values. Parameter with interference

Rate of change (%)

y11(s)+k y21(s)+k y31(s)+k

73.33 61.62 74.40

system as follow:

⎧ 3  ⎪ ⎪ y˜˙ r1 (t ) = Ir1 − dr1 y˜r1 (t ) + av1r1 (y˜r1 (t ) ) fv1 (y˜v1 (t ) ) ⎪ ⎪ ⎪ v=1,v=r ⎪ ⎪ 3 ⎪  ⎪ ⎪ + bv1r1 (y˜r1 (t ) ) fv1 (y˜v1 (t − τv1r1 (t ) ) ) ⎨ v=1,v=r

T  ⎪ ⎪ + cv1r1 (y˜r1 (t ) ) t −ρ (t ) fv1 (y˜v1 (s ) )ds ⎪ ⎪ v =1 , v  = r ⎪   ⎪ ⎪ ⎪ +ur1 (t ) + w1r1 ∗ pr1 (t ) − ηr1 (t ) − δ˜r1 (t ) , ⎪ ⎪ ⎩ ˜˙ δ r1 (t ) = w2r1 ∗ ( pr1 (t ) − ηr1 (t ) − δ˜r1 (t )). 3

and δ˜r1 (0 ) = 0, r = 1, 2, 3. The remaining settings, such as activation function, synchronization controller and so on, are the same as those in Example 1. Then, we have the corresponding response

(29)

W. Wang, X. Jia and X. Luo et al. / Chaos, Solitons and Fractals 126 (2019) 85–96

We set the plaintext signals to: p11 (t ) = 0.07sin(2t ) + 0.05, p21 (t ) = 0.05cos(1.5t ) + 0.06tanh(t ), p31 (t ) = (0.03cos(0.5t ) − 0.04tanh(t ))sin(t ), bring them into the above systems for encryption and decryption. Fig. 10 show that system (28) and (29) have great chaotic characteristics and they are very suitable for encryption. Fig. 11 depicts the change of plaintext signals pk1 (t), decrypted plaintext signals ρ k1 (t), and encrypted signals ωk1 (t) over time. It demonstrates that the effect of the security communication is reasonable. The security is central to the encryption algorithm. In the above encryption algorithm, controller is the key to decryption. The encryption will fail if the parameters are wrong, as shown in Fig. 12. From the perspective of secret key space, the secret key of this encryption scheme includes controller, drive-response systems, system parameters and initial values, obviously the key space is large enough to effectively resist the exhaustive attack. In addition, the chaotic sequence used in this project also makes the encryption scheme have initial value sensitivity, and the ciphertext sequence will produce more than 50% change if a small interference is applied to the encrypted chaotic sequence, as shown in Table 1, the small interference k = 10−10 . Remark 2. In Example 3, we use the chaos characteristics of system (28) and (29) with mixed delays to encrypt the plaintext signals. The plaintext signals are hidden in the chaotic sequence. Each neuron can transmit a signal, this improved the utilization of the networks effectively. Remark 3. The drive-response systems used in Example 3 considered mixed delays, this can reduce the adverse effects caused by the delay in the process of transmission. And the synchronization time is predictable, which makes the secret transmission more reliable. Remark 4. Driving system and response system are two interrelated networks. The faults of one node are likely to cause more problems, cascading failures will occur [33]. The synchronization effect will be affected, which leads to the failure of decryption. Therefore, the study of cascading failures is an outstanding challenge. This paper does not consider this problem, we will carry out related research in future research. 5. Conclusion In this paper, we discussed the fixed-time synchronization of MMAMNNs with distributed delay and time-varying delays. Some effective controllers were designed based on the multilayered structure and different activation functions. Then, a set of criteria about fixed-time synchronization have been constructed by Lyapunov stability theories. The settling time, which can be calculated in advance, is only related to the structure of MMAMNNs and the parameters of controllers but not related to the initial values. Using this advantage and the chaotic property of memristor, we designed a security communication scheme, compared with other synchronous control, the synchronization time can be more accurately grasped, thereby improving the decryption accuracy. Besides, in the final numerical simulations, we verified the correctness of the results, and the security of the encription scheme was proved. Declaration of Competing Interest We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled. All the authors

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Acknowledgments This work was supported by the National Key Research and Development Program of China under Grants 2017YFB0702300, the National Natural Science Foundation of China under Grants U1736117 and U1836106, the State Scholarship Fund of China Scholarship Council (CSC), the Fundamental Research Funds for the Central Universities under Grant 0650 0 025, the National Key Technologies R&D Program of China under Grant 2015BAK38B01 and the University of Science and Technology Beijing-National Taipei University of Technology Joint Research Program under Grant TW201705.

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