The existence and stability of the anti-periodic solution for delayed Cohen–Grossberg neural networks with impulsive effects

Neurocomputing 149 (2015) 22–28

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The existence and stability of the anti-periodic solution for delayed Cohen–Grossberg neural networks with impulsive effects Abdujelil Abdurahman, Haijun Jiang n College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046 Xinjiang, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 29 June 2013 Received in revised form 29 August 2013 Accepted 17 September 2013 Available online 31 July 2014

In this paper, we investigate the existence and global exponential stability of the anti-periodic solution for delayed Cohen–Grossberg neural networks with impulsive effects. First, based on the Lyapunov functional theory and by applying inequality technique, we give some new and useful sufficient conditions to ensure existence and exponential stability of the anti-periodic solutions. Then, we present an example with numerical simulations to illustrate our results. & 2014 Published by Elsevier B.V.

Keywords: Cohen–Grossberg neural network Anti-periodic solution Exponential stability Impulsive effect Time-varying delay

1. Introduction Since Cohen–Grossberg neural networks (CGNNs) have been first introduced by Cohen and Grossberg in 1983 [1], they have been intensively studied due to their promising potential applications in classification, parallel computation, associative memory and optimization problems. In these applications, the dynamics of networks such as the existence, uniqueness, Hopf bifurcation and global asymptotic stability or global exponential stability of the equilibrium point, periodic and almost periodic solutions for networks plays a key role, see [2–7] and references cited therein. Time delays unavoidably exist in the implementation of neural networks due to the finite speed of switching and transmission of signals. Besides delay effects, it has been observed that many evolutionary processes, including those related to neural networks, may exhibit impulsive effects. In these evolutionary processes, the solutions of system are not continuous but present jumps which could cause instability in the dynamical systems. Since the existence of delays and impulses is frequently a source of instability, bifurcation and chaos for neural networks, it is important to consider both delays and impulsive effects when investigating the stability of CGNNs, see [8–12]. Over the past decades, the anti-periodic solutions of Hopfield neural networks, recurrent neural networks and cellular neural networks have actively been investigated by a large number of scholars. For details, see [13–15,18] and references therein. In [13], the author considered the existence and exponential stability of the anti-periodic solutions for a class of recurrent neural networks with time-varying n

Corresponding author.

http://dx.doi.org/10.1016/j.neucom.2013.09.071 0925-2312/& 2014 Published by Elsevier B.V.

delays and continuously distributed delays. In [14], by constructing fundamental function sequences based on the solution of networks, the authors studied the existence of anti-periodic solutions for Hopfield neural networks with impulses. In [15], by establishing impulsive differential inequality and using Krasnoselski's fixed point theorem together with Lyapunov function method, the authors investigated the existence and exponential stability of anti-periodic solution for delayed cellular neural networks with impulsive effects. In [16], by constructing fundamental function sequences based on the solution of networks, the authors studied the existence and exponential stability of anti-periodic solutions for a class of delayed CGNNs. However, till now, there are very few or even no results on the problems of anti-periodic solutions for delayed CGNNs with impulsive effects, while the existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [17]). Thus, it is worth investigating the existence and stability of antiperiodic solutions for CGNNs with both time-varying delays and impulsive effects. Motivated by the above discussions, in this paper, we are concerned with the existence and the exponential stability of antiperiodic solutions for a class of impulsive CGNNs model with periodic coefficients and time-varying delays. By using analysis technique and constructing suitable Lyapunov function, we establish some simple and useful sufficient conditions on the existence and exponential stability of anti-periodic solutions for CGNNs with impulsive effects. The rest of the paper organized as follows. In Section 2, the proposed model is presented together with some related definitions. In addition, a preliminary lemma is given. Next section is devoted to investigate the existence and exponential stability of anti-periodic solution of the addressed networks. An illustrative example ends Section 4.

A. Abdurahman, H. Jiang / Neurocomputing 149 (2015) 22–28

23

that

2. Preliminaries

j f j ðuÞ  f j ðvÞj r L1j ju  vj;

In this paper, we consider the following impulsive CGNNs model with time-varying delays:

jg j ðuÞ g j ðvÞj r L2j ju  vj; u; v A R:

" # 8 n n > > < x0 ðtÞ ¼ ai ðxi ðtÞÞ  bi ðt; xi ðtÞÞ þ ∑ cij ðtÞf j ðxj ðtÞÞ þ ∑ dij ðtÞg j ðxj ðt  τij ðtÞÞÞ þ I i ðtÞ ; i j¼1

> > : Δxi ðt Þ ¼ xi ðt þ Þ  xi ðt Þ ¼ γ xi ðt Þ; k k k ik k

where n denotes the number of neurons in the network, xi(t) corresponds to the state of the ith unit at time t, ai ðxi ðtÞÞ represents an amplification function, bi ðt; xi ðtÞÞ is an appropriate behaved function, f j ðxj ðtÞÞ and g j ðxj ðt  τij ðtÞÞÞ denote, respectively, the measures of activation to its incoming potentials of the unit j at time t and t  τij ðtÞ, τij ðtÞ corresponds to the transmission delay along the axon of the jth unit and is non-negative function, and Ii(t) denotes the external bias on the ith unit at time t. Concerning coefficients of differential system (1), cij(t) denotes the synaptic connection weight of the unit j on the unit i at time t, dij(t) denotes the synaptic connection weight of the unit j on the unit i at time t  τij ðtÞ, where τij ðtÞ 4 0; i; j ¼ 1; 2; …; n. Throughout the paper, we always use i; j ¼ 1; 2; …; n, unless otherwise stated. The initial conditions associated with system (1) are given by xi ðsÞ ¼ φi ðsÞ;

s A ½  τ; 0;

where τ ¼ max1 r i;j r n fsupt A R þ τij ðtÞg; φi ðsÞ A Cð½  τ; 0; RÞ, which denotes the Banach space of all continuous functions mapping ½  τ; 0 into R with 1norm defined by ( ) J φ J 1 ¼ max

1rirn

sup jφi ðsÞj :

s A ½  τ;0

A solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT of impulsive system (1) is a piecewise continuous vector function whose components belong to the space PCð½  τ; þ 1Þ; RÞ ¼ fφðtÞ : ½  τ; þ 1Þ⟶R is continuous for t a t k ; φðt k Þ; φðt kþ Þ A R and φðt k Þ ¼ φðt k Þg: A function uðtÞ A PCð½  τ; þ 1Þ; RÞ is said to be ωanti  periodic, if (

uðt þ ωÞ ¼  uðtÞ;

t a tk ;

uððt k þ ωÞ þ Þ ¼ uðt kþ Þ;

k ¼ 1; 2; … :

Let Rn be n-dimensional vector space. For any u ¼ ðu1 ; u1 ; …; un ÞT A Rn , its norm is defined by

j¼1

In addition, we also formulate the following assumptions: þ

(H1) ai A CðR; R Þ and there exist positive constants ai and a i such that ai r ai ðuÞ r a i

for all u A R:

(H2) For each u, bi ð; uÞ is continuous, bðt; 0Þ  0 and there exists a continuous and ω-periodic function βi ðtÞ 4 0 such that bi ðt; uÞ  bi ðt; vÞ Z βi ðtÞ; uv

u; v A R; u a v:

(H3) The activation functions f j ðuÞ; g j ðuÞ are continuous, bounded and there exist Lipschitz constants L1j ; L2j 40 such

ð1Þ

k A N 9 f1; 2; …g;

(H4) cij ; dij ; τij ; I i A CðR; RÞ, and there exists a constant ω 4 0 such that ai ð uÞ ¼ ai ðuÞ; bi ðt þ ω; uÞ ¼  bi ðt; uÞ; cij ðt þ ωÞf j ðuÞ ¼  cij ðtÞf j ð  uÞ; I i ðt þ ωÞ ¼  I i ðtÞ; dij ðt þ ωÞg j ðuÞ ¼  dij ðtÞg j ð  uÞ;

τij ðt þ ωÞ ¼ τij ðtÞ:

(H5) j1 þ γ ik jr 1 and there exists a positive integer q such that ½0; ω \ ft k gk Z 1 ¼ ft 1 ; t 2 ; …; t q g; γ iðk þ qÞ ¼ γ ik ; k A N:

t k þ q ¼ t k þ ω;

About the jumps, ft k gk Z 1 is a strictly increasing sequence of positive numbers such that t k þ 1  t k Z κ, for all k A N; t k -1 as k-1, where κ 4 0. Δxi ðt k Þ ¼ xi ðt kþ Þ  xi ðt k Þ represents the abrupt change of xi(t) at impulsive moment t k . The following definition is given to obtain our results. Definition 1. Let xn ðtÞ ¼ ðxn1 ðtÞ; xn2 ðtÞ; …; xnn ðtÞÞT be an anti-periodic solution of differential system ð1Þ with initial value φn ¼ ðφn1 ðsÞ; φn2 ðsÞ; …; φnn ðsÞÞT . If there exist some constants λ 4 0 and M 41 such that for every solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT of system ð1Þ with any initial value φ ¼ ðφ1 ðsÞ; φ2 ðsÞ; …; φn ðsÞÞT , J xðtÞ xn ðtÞ J 1 r M J φ  φn J 1 e  λt ;

8 t Z 0;

n

where J φ φ J 1 ¼ sup  τ r s r 0 max1 r i r n jφi ðsÞ  φni ðsÞj. Then, the anti-periodic solution xn ðtÞ of system (1) is said to be globally exponentially stable. In addition, in the proof of the main results we shall need the following lemma. Lemma 1. Let hypotheses ðH1 Þ–ðH3 Þ and ðH5 Þ be satisfied and suppose there exist n positive constants p1 ; p2 ; …; pn such that n

n

j¼1

j¼1

 βi ðtÞpi þ ∑ jcij ðtÞjL1j pj þ ∑ jdij ðtÞjL2j pj þ Di ðtÞ o0;

J u J 1 ¼ max jui j: 1rirn

t Z 0; t a t k ;

t A ½0; ω;

ð2Þ

where n

n

j¼1

j¼1

Di ðtÞ ¼ ∑ jcij ðtÞjjf j ð0Þj þ ∑ jdij ðtÞjjg j ð0Þj þ I i ðtÞ: Then any solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT of system (1) with initial condition xi ðsÞ ¼ φi ðsÞ;

jφi ðsÞj o pi ;

t A ½  τ; 0; i ¼ 1; 2; …; n

verifies jxi ðsÞj opi

for all t 4 0; i ¼ 1; 2; …; n:

ð3Þ

Proof. For any assigned initial condition, hypotheses ðH1 Þ, ðH2 Þ and ðH3 Þ guarantee the existence and uniqueness of x(t), the

24

A. Abdurahman, H. Jiang / Neurocomputing 149 (2015) 22–28

solution to (1) in ½0; þ 1Þ. Assume that the initial condition verifies jφi ðsÞj o pi ;

s A ½  τ; 0; i ¼ 1; 2; …; n:

t A ð0; t 1 :

ð4Þ

D þ jxi ðtÞjt ¼ σ Z 0

and

jxj ðtÞj o pj ; t A ½  τ; σÞ; j ¼ 1; 2; …; n;

ð5Þ þ

where D denotes Dini upper left derivative. From ðH1 Þ–ðH3 Þ, (2) and (5), we get " n

0 r ai ðσÞ  βðσÞjxi ðσÞj þ ∑ jcij ðσÞjjf j ðxj ðσÞÞj j¼1

j¼1

n

j¼1

j¼1

n

j¼1

ð10Þ Also

Z 

 ui ðt kþ Þ r  

xi ðt kþ Þ xni ðt kþ Þ

 Z  ds   ð1 þ γ ik Þxi ðt k Þ ds  ¼ : ai ðsÞ  ð1 þ γ ik Þxn ðt k Þ ai ðsÞ i

Making the substitution θ ¼ ð1 þ γ ik Þ=s, we get Z   xi ðtk Þ ð1 þ γ Þ  a   ik dθ r i j1 þ γ ik jui ðt k Þ: ui ðt kþ Þ r   xn ðtk Þ ai ðð1 þ γ ik ÞθÞ  ai

ð11Þ

Considering the Lyapunov function

þ ∑ jdij ðσÞjjg j ðxj ðσ τij ðσÞÞÞj þ jI i ðσÞj n

n

j¼1

u0i ðtÞ r  βi ðtÞai ui ðtÞ þ ∑ jcij ðtÞjL1j a j uj ðtÞÞ þ ∑ jdij ðtÞjL2j a j uj ðt  τij ðtÞÞÞ:

i

#

n

"

 f j ðxj ðtÞÞÞ þ ∑ dij ðtÞðg j ðxj ðt  τij ðtÞÞÞ  g j ðxj ðt  τij ðtÞÞÞÞ : From (H2), (H3) and (9), it follows that

In fact, if it does not hold, then there exist some i and time σ A ð0; t 1  such that jxi ðσÞj ¼ pi ;

n

j¼1

Now, we first claim that, for each i ¼ 1; 2; …; n, jxi ðtÞj o pi ;

#

n

n

V i ðtÞ ¼ ui ðtÞeλt ;

#

i ¼ 1; 2; …; n;

ð12Þ

and for t a t k calculating the upper left derivative of Vi(t) along the trajectory of system (1), (10) leads to

r ai ðσÞ βi ðσÞpi þ ∑ jcij ðσÞjL1j pj þ ∑ jdij ðσÞjL2j pj þ Di ðσÞ o 0;

D þ V i ðtÞ ¼ λeλt ui ðtÞ þ eλt u0i ðtÞ n

which is a contradiction and shows that (4) is true for t A ð0; t 1 . Since jxi ðt 1þ Þj ¼ j1 þ γ i1 jjxi ðt 1 Þj r jxi ðt 1 Þj, using the same method we can prove that jxi ðtÞj o pi ;

r ðλ  βi ðtÞai Þeλt ui ðtÞ þ ∑ jcij ðtÞjL1j a j eλt uj ðtÞÞ j¼1

n

þ ∑ jdij ðtÞjL2j a j eλt uj ðt  τij ðtÞÞÞ

t A ðt 1 ; t 2 ; i ¼ 1; 2; …; n;

j¼1

and so on. The proof of Lemma 1 is now completed.

n

□

r ðλ  βi ðtÞai ÞV i ðtÞ þ ∑ jcij ðtÞjL1j a j V j ðtÞ j¼1

n

3. Main results

þ ∑

j¼1

In this section, we consider the existence and global exponential stability of anti-periodic solutions for system ð1Þ. Suppose that xn ðtÞ ¼ ðxn1 ðtÞ; xn2 ðtÞ; …; xnn ðtÞÞT is a solution of system ð1Þ with initial conditions xni ðsÞ ¼ φni ðsÞ;

jφni ðsÞj o pi ; s A ½ τ; 0;

ð6Þ

where pi are defined in Lemma 1. Then, on the stability of system ð1Þ, we have the following results. Theorem 1. Let ðH1 Þ–ðH3 Þ and ðH5 Þ hold. Assume that the following inequality is satisfied: n

n

j¼1

j¼1

ðλ  βi ðtÞÞpi þ ∑ jcij ðtÞjL1j pj þ ∑ jdij ðtÞjL2j pj eλτ o0;

t A ½0; ω;

ð7Þ

where λ is a positive constant such that ðλ  ln A=κÞ 4 0, A 9 max1 r i r n a i =ai . Then, the anti-periodic solution xn ðtÞ of system (1) is exponentially stable. Proof. Let xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT be an arbitrary solutions of system (1) with initial value φ ¼ ðφ1 ðsÞ; φ2 ðsÞ; …; φn ðsÞÞT and set Z xi ðtÞ ds : ð8Þ ui ðtÞ ¼ signðxi ðtÞ  xni ðtÞÞ xn ðtÞ ai ðsÞ i

Let m 4 1 be an arbitrary real number such that jφi ðsÞ φni ðsÞj ; ai s A ½  τ;01 r i r n

mpi 4 sup

max

i ¼ 1; 2; …; n:

It follows from (9) and (12) that V i ðtÞ ¼ ui ðtÞeλt o mpi ;

t A ½  τ; 0; i ¼ 1; 2; …; n:

Now, we claim that V i ðtÞ ¼ ui ðtÞeλt o Ak mpi ;

t A ðt k ; t k þ 1 ; k A N; i ¼ 1; 2; …; n;

ð14Þ

where A ¼ max1 r i r n a i =ai . First, we prove that V i ðtÞ ¼ ui ðtÞeλt o mpi ;

i ¼ 1; 2; …; n;

is true for 0 o t r t 1 . Otherwise, there exist i A f1; 2; …; ng and η A ð0; t 1  such that V i ðηÞ ¼ mpi

and

V j ðtÞ ompj ; t A ½  τ; ηÞ; j ¼ 1; 2; …; n:

ð15Þ

Combining (7), (13) and (15), we get n

jxi ðtÞ  xni ðtÞj jxi ðtÞ  xni ðtÞj r ui ðtÞ r : ai ai

u0i ðtÞ ¼

ð13Þ

0 r D þ V i ðtÞjt ¼ η

From (H1), it is easy to check that

For t a t k , we have

jdij ðtÞjL2j a j eλτ V j ðt  τij ðtÞÞ:

ð9Þ

n

rðλ  ai ðηÞÞV i ðηÞ þ ∑ jbij ðηÞjL1j V j ðηÞ þ ∑ jcij ðηÞjL2j V j ðη τij ðηÞÞeλτ "

j¼1

j¼1

n

n

j¼1

j¼1

#

rm ðλ  ai ðηÞÞpi þ ∑ jbij ðηÞjL1j pj þ ∑ jcij ðηÞjL2j pj eλτ o 0; 

 x0i ðtÞ xn 0i ðtÞ  signðxi ðtÞ  xi ðtÞÞ ai ðxi ðtÞÞ ai ðxni ðtÞÞ " n

n

¼ signðxi ðtÞ  xni ðtÞÞ  ðbi ðt; xi ðtÞÞ bi ðt; xni ðtÞÞÞ þ ∑ cij ðtÞðf j ðxj ðtÞÞ j¼1

which is a contradiction. Hence (14) holds for t A ½  τ; t 1 . From (11) and j1 þ γ ik j r 1, we know that a V i ðt 1þ Þ r i j1 þ γ i1 ðt 1 ÞjV i ðt 1 Þ r AV i ðt 1 Þ rV i ðt 1þ Þ rAmpi : ai

A. Abdurahman, H. Jiang / Neurocomputing 149 (2015) 22–28

Thus, for any natural number h; xi ðt þ ðh þ 1ÞωÞ are the solutions of system (1). Then, by Theorem 1, there exist constants M 41 and λ 4 0, such that

By the same way, we can prove that λt

V i ðtÞ ¼ ui ðtÞe o Ampi ;

t A ðt 1 ; t 2 ; i ¼ 1; 2; …; n:

Repeating the above process, when t A ðt k ; t k þ 1 , we get V i ðtÞ ¼ ui ðtÞeλt o Ak mpi ;

jð  1Þh þ 1 xi ðt þðh þ 1ÞωÞ  ð  1Þh xi ðt þhωÞj

t A ðt k ; t k þ 1 ; i ¼ 1; 2; …; n:

r Me  λðt þ hωÞ sup jxi ðs þ ωÞ þxi ðsÞj

Since Δt k ¼ t k þ 1  t k Z κ and A 41, when t A ðt k ; t k þ 1 , we get k

k ln A

A ¼e

½1 þ ððt 2  t 1 Þ þ ðt 3  t 2 Þ þ ⋯ þ ðt k  t k  1 ÞÞ=κln A

rM e

0

s A ½  τ;0

r 2Mpi e  λðt þ hωÞ

ð1 þ ðt k  t 1 Þ=κÞln A

re

0 ðt=κÞln A

25

¼e

;

ð22Þ

and

ð1 þ t 1 =κÞln A

where M ¼ e

. From (9) and the above inequality, we have 0

n

jxi ðtÞ  xi ðtÞj r a i ui ðtÞ r ma i M pi e

 ðλ  ln A=κÞt

;

jð  1Þh þ 1 xi ððt k þ ðh þ 1ÞωÞ þ Þ  ð  1Þh xi ððt k þ hωÞ þ Þj ¼ jxi ððt k þ ðhþ 1ÞωÞ þ Þ þ xi ððt k þ hωÞ þ Þj

8 t 4 0; i ¼ 1; 2; …; n:

¼ j1 þγ ik jxi ðt k þ ðhþ 1ÞωÞ þ xi ðt k þ hωÞj r 2Mpi e  λðtk þ hωÞ ;

ð16Þ

8 k A N: ð23Þ

Letting M 41 such that 0

max fma i M pi g r M J φ  φn J 1 ;

1rirn

i ¼ 1; 2; …; n:

ð17Þ

m

ð  1Þm þ 1 xi ðt k þ ðm þ 1ÞωÞ ¼ xi ðtÞ þ ∑ ð 1Þl þ 1 ½xi ðt þ ðl þ1ÞωÞ

Together with (9), (16) and (17), we get J xðtÞ  xn ðtÞ J 1 ¼ max jxi ðtÞ  xn ðtÞj rM J φ  φn J 1 e  δt ; 1rirn

l¼0

8 t 4 0;

 ð  1Þl xi ðt þ lωÞ:

ð18Þ where δ ¼ ðλ  ln A=κÞ 40. Therefore, according to Definition 1, xn ðtÞ is globally exponentially stable. The proof of Theorem 1 is completed. □ The following theorem is provided to guarantee the existence of anti-periodic solution of system (1). Theorem 2. Suppose that hypotheses ðH1 Þ–ðH5 Þ hold. If the inequality (2) is satisfied, then system (1) admits an ω-anti-periodic solution. Proof. Let xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; …; xn ðtÞÞT be a solution of system ð1Þ with initial conditions xi ðsÞ ¼ φi ðsÞ;

jφi ðsÞj r pi ; s A ½  τ; 0; i ¼ 1; 2; …; n;

ð24Þ

Then 8 m mþ1 > > xi ðt þ ðmþ 1ÞωÞj rjxi ðtÞj þ ∑ ð  1Þl þ 1 jxi ðt þ ðl þ1ÞωÞ > jð  1Þ > > l¼0 > <  ð  1Þl xi ðt þlωÞj; t at k ; > > mþ1 þ > jð  1Þ xi ððt k þ ðmþ 1ÞωÞ Þj rjð1 þ γ ik Þð  1Þm þ 1 xi ðt k þ ðm þ 1ÞωÞj > > > : r jxi ðt k þðmþ 1ÞωÞj:

ð25Þ 0

In view of (22), we can choose a sufficiently large constant N and a positive constant ε such that jxi ðt þ ðl þ1ÞωÞ  ð  1Þl xi ðt þ lωÞj r εðe  λω Þl ;

l 4 N0 ;

ð26Þ

þ

where pi are defined in (2). By Lemma 1, we have jxi ðtÞj r pi ;

Thus, when t a t k , for any natural number m, we obtain

8 t A ½  τ; 1Þ; i ¼ 1; 2; …; n:

ð19Þ

From hypotheses (H4), for any hA N, we have ½ð  1Þh þ 1 xi ðt þ ðh þ 1ÞωÞ0 ¼ ð  1Þh þ 1 x0i ðt þðh þ 1ÞωÞ " ( ¼ ð  1Þh þ 1 ai ðxi ðt þ ðh þ 1ÞωÞÞ  bi ðt þ ðh þ 1Þω; xi ðt þ ðh þ1ÞωÞÞ

on any compact set of R . Together with (25) and (26), it follows that fundamental sequence fð  1Þm þ 1 xi ðt þ ðm þ 1ÞωÞg uniformly converges to a piecewise continuous function xni ðtÞ on any compact set of R þ . Now we will show that xn ðtÞ ¼ ðxn1 ðtÞ; xn2 ðtÞ; …; xnn ðtÞÞT is an ωanti-periodic solution of system (1). It is easily known that xn ðtÞ is anti-periodic, since xni ðt þ ωÞ ¼ lim ð  1Þm xi ðt þ ωþ mωÞ m- þ 1

n

¼

þ ∑ cij ðt þðh þ 1ÞωÞf j ðxj ðt þ ðh þ 1ÞωÞÞ þ I i ðt þ ðh þ 1ÞωÞ j¼1

#)

n

( ¼

for t at k ;

and

þ ∑ dij ðt þ ðh þ1ÞωÞg j ðxj ðt þ ðh þ 1Þω τij ðt þ ðh þ1ÞωÞÞÞ j¼1

ð  1Þm þ 1 xi ðt þ ðm þ 1ÞωÞ ¼ xni ðtÞ

lim

ðm þ 1Þ- þ 1

xni ððt k þ ωÞ þ Þ ¼ lim ð  1Þm xi ððt k þ ω þmωÞ þ Þ

"

m- þ 1

¼

ai ðð  1Þh þ 1 xi ðt þ ðh þ 1ÞωÞÞ  bi ðt; ð  1Þh þ 1 xi ðt þðh þ 1ÞωÞÞ

lim

ðm þ 1Þ- þ 1

ð  1Þm þ 1 xi ððt þ ðm þ 1ÞωÞ þ Þ ¼  xni ðt kþ Þ:

n

þ ∑ cij ðtÞf j ðð  1Þh þ 1 xj ðt þ ðh þ 1ÞωÞÞ þ I i ðtÞ j¼1

#)

n

þ ∑ dij ðtÞg j ðð  1Þh þ 1 xj ðt þ ðh þ 1Þω τij ðtÞÞÞ j¼1

¼

> > : xn ðt þ Þ

¼

k

ð20Þ "

8 > > < xn 0i ðtÞ i

:

n

n

Next, we prove that xn ðtÞ is a solution of system (1). In fact, together with the piece-wise continuity of the right side of (1), (20) and (21) imply that fð  1Þm þ 1 xi ððt þ ðm þ 1ÞωÞÞg0 converges uniformly to a piece-wise continuous function on any compact n

n

# n

ai ðxi ðtÞÞ bi ðt; xi ðtÞÞ þ ∑ cij ðtÞf j ðxj ðtÞÞ þ ∑ dij ðtÞg j ðxj ðt  τij ðtÞÞÞþ I i ðtÞ ; j¼1

ð1 þ γ ik Þxni ðt k Þ;

j¼1

t a tk ;

k ¼ 1; 2; … :

Moreover, by hypothesis of (H5), we have ð  1Þh þ 1 xi ððt k þðh þ 1ÞωÞ þ Þ ¼ ð  1Þh þ 1 ð1 þ γ iðk þ ðh þ 1ÞqÞ Þxi ðt k þ ðh þ 1ÞωÞ ¼ ð1 þ γ ik Þ½ð  1Þh þ 1 xi ðt k þðh þ 1ÞωÞ:

n

ð21Þ

set of R þ . Therefore, letting m- þ1 on both sides of (20) and (21), we obtain Thus, xn ðtÞ ¼ ðxn1 ðtÞ; xn2 ðtÞ; …; xnn ðtÞÞT is an ω-anti-periodic solution of system (1). The proof of Theorem 2 is now completed. □

26

A. Abdurahman, H. Jiang / Neurocomputing 149 (2015) 22–28

Suppose that xn ðtÞ is an ω-anti-periodic solution of system (1), then it is not difficult to know that xn ðtÞ is also 2ω-periodic solution of system (1). Thus from Theorem 2, we have the following result.

and

Corollary 1. Assume that hypotheses ðH1 Þ–ðH5 Þ and inequality (2) are all satisfied, then system (1) admits a 2ω-periodic solution.

a2 ðuÞ ¼ 1þ

Remark 1. Let ðH1 Þ–ðH5 Þ hold and suppose that inequalities (2) and (7) are both satisfied. Then it follows from Theorems 1, 2 and Definition 1 that system (1) has an ω-anti-periodic solution xn ðtÞ which is globally exponentially stable. Remark 2. Because the inequalities (2) and (7) are both timevarying, it is not easy to find suitable p1 ; p2 ; …; pn , which satisfy inequalities (2) and (7). However, if we let p1 ¼ p2 ¼ ⋯ ¼ pn 9 p and assume that the following inequality holds: n

j¼1

j¼1

t A ½0; ω;

then inequality (2) will become " n

n

j¼1

j¼1

ð27Þ

ðdij Þ22 ¼

I 1 ðtÞ ¼ 3:2 cos ðπtÞ;

b1 ðt; uÞ ¼ ð6:1  j cos ðπtÞjÞu;    π  b2 ðt; uÞ ¼ 4:9þ  sin πt þ jÞu; 3

 π I2 ðtÞ ¼ 1:9 sin πt þ ; 3

0:9j sin ðπtÞj

0:35j cos ðπtÞj

j cos ðπtÞj

j cos ðπtÞj

0:7j cos ðπtÞj

0:5j sin ðπtÞj

j cos ðπtÞj

j cos ðπtÞj

! ;

! :

It is easy to check that ai ðuÞ; f i ðuÞ; g i ðuÞ satisfy hypotheses (H1) and (H3) with a 1 ¼ 2; a1 ¼ 1:5; a 2 ¼ 1:5; a2 ¼ 1; L1i ¼ L2i ¼ 1; i ¼ 1; 2. If we let β1 ðtÞ ¼ ð6:1 j cos ðπtÞjÞ; β2 ðtÞ ¼ ð4:9 þ j sin ðπt þ π=3ÞjÞ, then hypothesis (H2) is also satisfied. About the jumps, we suppose t 1 ¼ 0:2;

t 2 ¼ 1:2;

t 3 ¼ 2:2; … :

So that, in interval ½0; 1, we have only one jump. Let

#1

p 4 Di ðtÞ βi ðtÞ  ∑ jcij ðtÞjL1j  ∑ jdij ðtÞjL2j

0:5 ; 1 þ u2

0:5 ; 1 þ u2

ðcij Þ22 ¼

;

2

t A ½0; ω:

2

χ i ðtÞ ¼ βi ðtÞpi  ∑ jcij ðtÞjL1j pj  ∑ jdij ðtÞjL2j pj  jDi ðtÞj; j¼1

Thus, if we take sufficiently large p, then the inequality (2) is always satisfied. In addition, when the inequality (27) holds, it is not difficult to know that the inequality (7) is also satisfied for small enough λ. Therefore, to check whether inequalities (2) and (7) are satisfied or not, we only need to check the inequality (27). Remark 3. In this paper, for the first time, we investigate the existence and global exponential stability of anti-periodic solution for a class of impulsive CGNNs with time-varying delays. It is known that CGNNs model (1) includes some well-known neural networks such as Hopfield neural networks, cellular neural networks and recurrent neural networks as a special case. From this point, we can conclude that our results are more practical than those in [13–15].

j¼1

2

2

j¼1

j¼1

ψ i ðλ; tÞ ¼ ðλ  βi ðtÞÞpi þ ∑ jcij ðtÞjL1j pj þ ∑ jdij ðtÞjL2j pj eλτ ;

i ¼ 1; 2:

If we take p1 ¼ p2 ¼ 1, and λ ¼ ln A=κ þ 0:01, where A ¼ 1:5; κ ¼ 1, then it is not difficult to check that minðχ 1 ðtÞÞ ¼ 0:6665,

0.8 0.6 0.4 0.2

x1

n

βi ðtÞ  ∑ jcij ðtÞjL1j  ∑ jdij ðtÞjL2j 4 0;

a1 ðuÞ ¼ 2 

0

Remark 4. If we let dij ðtÞ ¼ 0 or g j ðxÞ ¼ 0, and ai ðuÞ  1; bi ðt; uÞ  bi ðuÞ, then system (1) becomes 8 n > < x0i ðtÞ ¼  bi ðtÞxi ðtÞ þ ∑ cij ðtÞf j ðxj ðtÞÞ þI i ðtÞ; t Z 0; t a t k ;

−0.2

> : x ðt þ Þ ¼ ð1 þγ Þx ðt Þ; i k ik i k

−0.8

−0.4 −0.6

j¼1

i ¼ 1; 2; …; n; k A N;

0

5

10

15

20

t

which is studied in [14]. It is not difficult to see that Theorem 2 includes the main results in [14] as a special case.

Fig. 1. Time response of state variable x1 with impulsive effects.

4. Numerical simulations 0.5

In this section, an example is given to illustrate the effectiveness of our results obtained in this paper. For n ¼2, consider the following delayed Cohen–Grossberg neural networks system:

ð28Þ

0.2 0.1 0

−0.1 −0.2 −0.3

with impulses Δx1 ðt k Þ ¼ 0:6x1 ðt k Þ;

Δx2 ðt k Þ ¼ 0:4x2 ; k ¼ 1; 2; …;

where f i ðuÞ ¼ tanhðu=2Þ;

0.3

x2

8 " # 2 2 > > < x0 ðtÞ ¼ a ðx1 ðtÞÞ  b ðt; x ðtÞÞ þ ∑ b ðtÞf ðx ðtÞÞ þ ∑ c ðtÞg ðx ðt  τ ðtÞÞÞ þ I ðtÞ ; i i i ij ij ij i j j j j i j¼1 j¼1 > > : t Z 0; t at ; i ¼ 1; 2; k

0.4

−0.4 −0.5 0

ðju þ 1j  ju  1jÞ ; g i ðuÞ ¼ 2

5

10

15

t

τi1 ðtÞ ¼ τi2 ðtÞ ¼ 1; i ¼ 1; 2

Fig. 2. Time response of state variable x2 with impulsive effects.

20

A. Abdurahman, H. Jiang / Neurocomputing 149 (2015) 22–28

0.6

Acknowledgements

0.4

This work was supported by the National Natural Science Foundation of PR China (Grant no. 61164004), the Natural Science Foundation of Xinjiang (Grant no. 2010211A07) and the Excellent Doctor Innovation Program of Xinjiang University (Grant no. XJUBSCX-2010003). The authors are grateful to the Editor and anonymous reviewers for their kind help and constructive comments.

0.2

x2

27

0

−0.2 −0.4

References

−0.6 −0.8

−0.6

−0.4

−0.2

0

x1

0.2

0.4

0.6

0.8

Fig. 3. Phase response of state variables x1 and x2 with different initial values.

0.5 0.4 0.3 0.2

x2

0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x1 Fig. 4. Phase response of anti-periodic solution of system (28).

minðχ 2 ðtÞÞ ¼ 0:0994, max ψ 1 ðλ; tÞ ¼  2:7591; max ψ 2 ðλ; tÞ ¼  0:2905. Thus, inequalities (2) and (7) are both satisfied. Therefore, by Theorems 1 and 2, the impulsive system (28) has a unique 1-antiperiodic solution which is globally exponentially stable. The fact can be seen by the numerical simulation in Figs. 1–4.

Remark 5. One can observe that all the results in [13–16] and the references therein cannot be applicable to system (28) to obtain the existence and exponential stability of the anti-periodic solutions. This implies that the results of this paper are essentially new.

[1] M.A. Cohen, S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern. 13 (1983) 815–826. [2] S. Arik, Z. Orman, Global stability analysis of Cohen–Grossberg neural networks with time varying delays, Phys. Lett. A 341 (2005) 410–421. [3] J. Cao, Q. Song, Stability in Cohen–Grossberg-type bidirectional associative memory neural networks with time-varying delays, Nonlinearity 19 (2006) 1601–1617. [4] Q. Song, J. Cao, Stability analysis of Cohen–Grossberg neural network with both time-varying and continuously distributed delays, J. Comput. Appl. Math. 197 (2006) 188–203. [5] H. Zhao, L. Wang, Stability and bifurcation for discrete-time Cohen–Grossberg neural network, Appl. Math. Comput. 179 (2006) 787–798. [6] H. Jiang, J. Cao, Z. Teng, Dynamics of Cohen–Grossberg neural networks with time-varying delays, Phys. Lett. A 354 (2006) 414–422. [7] H. Xiang, J. Cao, Almost periodic solution of Cohen–Grossberg neural networks with bounded and unbounded delays, Nonlinear Anal. RWA 10 (2009) 2407–2419. [8] Z. Wen, J. Sun, Stability analysis of delayed Cohen–Grossberg BAM neural networks with impulses via nonsmooth analysis, Chaos Solitons Fractals 42 (2009) 1829–1837. [9] X. Yang, Existence and global exponential stability of periodic solution for Cohen–Grossberg shunting inhibitory cellular neural networks with delays and impulses, Neurocomputing 72 (2009) 2219–2226. [10] K. Li, Stability analysis for impulsive Cohen–Grossberg neural networks with time-varying delays and distributed delays, Nonlinear Anal. RWA 10 (2009) 2784–2798. [11] C. Bai, Global exponential stability and existence of periodic solution of Cohen–Grossberg type neural networks with delays and impulses, Nonlinear Anal. RWA 9 (2008) 747–761. [12] J. Li, J. Yan, X. Jia, Dynamical analysis of Cohen–Grossberg neural networks with time-varying delays and impulses, Comput. Math. Appl. 58 (2009) 1142–1151. [13] J. Shao, An anti-periodic solution for a class of recurrent neural networks, J. Comput. Appl. Math. 228 (2009) 231–237. [14] P. Shi, L. Dong, Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses, Appl. Math. Comput. 216 (2010) 623–630. [15] L. Pan, J. Cao, Anti-periodic solution for delayed cellular neural networks with impulsive effects, Nonlinear Anal. RWA 12 (2011) 3014–3027. [16] Y. Li, L. Yang, Anti-periodic solutions for Cohen–Grossberg neural networks with bounded and unbounded delays, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3134–3140. [17] Y. Chen, J. Nieto, D. Oregan, Anti-periodic solutions for fully nonlinear first-order differential equations, Math. Comput. Model. 46 (2007) 1183–1190. [18] X. Chen, Q. Song, Global exponential stability of antiperiodic solutions for discrete-time neural networks with mixed delays and impulses, Discrete Dyn. Nat. Soc. 2012 (2012) 168375.

5. Conclusion In this paper, we study the existence and global exponential stability of the anti-periodic solution for impulsive Cohen–Grossberg neural networks with periodic coefficients and time-varying delays. Based on the Lyapunov functional theory, we give some new and useful sufficient conditions for the existence and exponential stability of the anti-periodic solutions by applying mathematical induction and inequality technique. Finally, we give an example with its numerical simulations to verify the effectiveness and feasibility of our results.

Abdujelil Abdurahman was born in Xinjiang Uyghur Autonomous Region, China, in 1987. He received the B.S. degree in mathematics and applied mathematics from the College of Mathematics and System Sciences, Xinjiang University, Urumqi, China, in 2010. In July 2013, he received the M.S. degree in operations research and control theory from Xinjiang University. Currently, he is working towards the Ph.D. degree at Xinjiang University. His current research interests include chaotic systems, neural networks, complex networks, and control theory.

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A. Abdurahman, H. Jiang / Neurocomputing 149 (2015) 22–28

Haijun Jiang was born in Hunan, China, in 1968. He received the B.S. degree from the Mathematics Department, Yili Teacher College, Yili, Xinjiang, China; the M.S. degree from the Mathematics Department, East China Normal University, Shanghai, China, and the Ph.D. degree from the College of Mathematics and System Sciences, Xinjiang University, China, in 1990, 1994, and 2004, respectively. He was a Post doctoral Research Fellow in the Department of Southeast University, Nanjing, China, from 2004.9 to 2006.9. He is a professor and Doctoral Advisor of Mathematics and System Sciences of Xinjiang University, Xinjiang, China. His current research interests include nonlinear dynamics, delay differential equations, dynamics of neural networks, mathematical biology.