numerical simulation of periodic solutions for a class of numerical discretization neural networks

Mathematical and Computer Modelling 52 (2010) 386–396

Contents lists available at ScienceDirect

Mathematical and Computer Modelling

journal homepage: www.elsevier.com/locate/mcm

Numerical simulation of periodic solutions for a class of numericaldiscretization neural networksI

Chun Lu a,∗, Xiaohua Ding b, Mingzhu Liu ca School of Science, Qingdao Technological University, Qingdao 266520, Chinab Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, Chinac Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

a r t i c l e i n f o

Article history:Received 28 May 2009Received in revised form 6 March 2010Accepted 8 March 2010

Keywords:Neural networkPeriodic solutionGlobally exponentially stableCoincidence degreeLyapunov function

a b s t r a c t

In this paper, the existence and global exponential stability of periodic solutions fora class of numerical discretization neural networks are investigated. Using coincidencedegree theory and the Lyapunov method, sufficient conditions for the existence and globalexponential stability of periodic solutions are obtained. Numerical simulations are given atthe end of this paper.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we consider the existence and stability of a numerical discretization neural network resulting from thesemi-discretization technique for neural networks with finite delays and distributed delays:

xi(t) = −bi(t)xi(t)+ fi

(t, x1(t), . . . , xm(t); x1(t − τi1(t)), . . . , xm(t − τim(t));

∫+∞

0ki1(s)x1(t − s)ds, . . . ,

∫+∞

0kim(s)xm(t − s)ds

)+ Ii(t), i = 1, . . . ,m. (1.1)

System (1.1) is a more general form of neural networks with delay. Many authors have discussed other neural networkswith finite delays [1–8], and most of their systems can be deduced from (1.1).For system (1.1) we make the following assumptions:(H1): For each i ∈ 1, . . . ,m, bi(t) is bounded and continuous with bi(t) > 0 and bi(t + ω) = bi(t) for t ∈ R+, Ii(t) is

bounded, continuous and Ii(t + ω) = Ii(t) for t ∈ R+, where ω is a positive constant.(H2): For each i, j ∈ 1, . . . ,m, kij : R+ → R+ satisfies:∫

+∞

0kij(s)ds = 1,

∫+∞

0skij(s)ds <∞.

I This paper is supported by the National Natural Science Foundation of China (10671047) and the foundation of HITC (200713).∗ Corresponding author. Tel.: +86 13864220728.E-mail address:[email protected] (C. Lu).

0895-7177/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2010.03.005

http://www.elsevier.com/locate/mcm

http://www.elsevier.com/locate/mcm

mailto:[email protected]

http://dx.doi.org/10.1016/j.mcm.2010.03.005

C. Lu et al. / Mathematical and Computer Modelling 52 (2010) 386–396 387

(H3): For each i ∈ 1, . . . ,m, fi : R+ × Rm × Rm × Rm → R is bounded, continuous and there exists nonnegative,bounded and continuous functions αij(t), βij(t), γij(t) defined on R+ such that

|fi(t, u1, . . . , um; v1, . . . , vm;w1, . . . , wm)− fi(t, u1, . . . , um; v1, . . . , vm; w1, . . . , wm)|

≤

m∑j=1

[αij(t)|uj − uj| + βij(t)|vj − vj| + γij(t)|wj − wj|

], i = 1, . . . ,m

for any (u1, . . . , um), (u1, . . . , um), (v1, . . . , vm), (v1, . . . , vm), (w1, . . . , wm), (w1, . . . , wm) ∈ Rm.For system (1.1), we consider initial conditions of the form

xi(t) = ϕi(t), t ∈ (−∞, 0], i = 1, . . . ,m (1.2)

where ϕi(t) is bounded and continuous on (−∞, 0].For convenience in our study, let Z denote the set of all integers; Z+0 = 0, 1, 2, . . .; [a, b]Z = a, a + 1, . . . , b − 1, b

where a, b ∈ Z, a ≤ b; and [a,+∞)Z = a, a + 1, a + 2, . . . where a ∈ Z. We begin approximating the continuous-timenetwork (1.2) by replacing the integral terms with discrete sums of the form

∫+∞

0kij(s)xj(t − s)ds ≈

+∞∑[ sh ]=1

ωij(h)kij([ sh

]h)xj

([th

]h−

[ sh

]h)

(1.3)

for t ∈ [nh, (n+1)h), s ∈ [ph, (p+1)h), n ∈ Z+0 , p ∈ Z+, where [r]denotes the integer part of a real number r ,h > 0 is a fixednumber denoting a uniform discretization step size satisfying h = ω

ω, ω ∈ N , ωij(h) > 0 for h > 0 and ωij(h) ≈ h + O(h2)

for small h > 0. We note that ωij(h) are chosen so that the analogue kernels

Hij

([ sh

]h)= ωij(h)kij

([ sh

]h)

(1.4)

satisfy the following properties:

(H4) : Hij : Z+ −→ [0,+∞),+∞∑p=1

Hij(p) = 1,+∞∑p=1

Hij(p)p < +∞.

This analogue has been employed elsewhere (see, for instance [9–12] in the formulation of discrete-time analogues of thedistributed delay). With (1.3) and (1.4), we approximate (1.1) by differential equations with piecewise constant argumentsof the form

xi(t) = −bi

([th

]h)xi(t)+ fi

([th

]h, x1

([th

]h), . . . , xm

([th

]h); x1

([th

]h

− τi1

([th

]h))

, . . . , xm

([th

]h− τim

([th

]h));

+∞∑[ sh ]=1

Hi1

([ sh

]h)

× x1

([th

]h−

[ sh

]h), . . . ,

+∞∑[ sh ]=1

Him

([ sh

]h)xm

([th

]h−

[ sh

]h))+ Ii

([th

]h)

(1.5)

for t ∈ [nh, (n + 1)h), s ∈ [ph, (p + 1)h), n ∈ Z+0 , p ∈ Z+. Noting that[ th

]= n,

[ sh

]= p and adopting the notation

u(n) = u(nh), we rewrite (1.5) as

xi(t) = −bi(n)xi(t)+ fi

(n, x1(n), . . . , xm(n); x1(n− τi1(n)), . . . , xm(n− τim(n));

+∞∑p=1

Hi1(p)x1(n− p), . . . ,+∞∑p=1

Him(p)xm(n− p)

)+ Ii(n), t ≥ 0, i = 1, . . . ,m. (1.6)

388 C. Lu et al. / Mathematical and Computer Modelling 52 (2010) 386–396

Integrating (1.6) over the interval [nh, t), where t < (n+ 1)h, we get

xi(t)ebi(n)t − xi(n)ebi(n)nh =ebi(n)t − ebi(n)nh

bi(n)

fi


+∞∑p=1

Hi1(p)x1(n− p), . . . ,+∞∑p=1

Him(p)xm(n− p)

)+ Ii(n)

, i = 1, . . . ,m (1.7)

and by allowing t → (n+ 1)h above, we obtain

xi(n+ 1) = xi(n)e−bi(n)h + θi(h)

(fi


+∞∑p=1

Hi1(p)x1(n− p), . . . ,+∞∑p=1

Him(p)xm(n− p)

)+ Ii(n)

), i = 1, . . . ,m, n ∈ Z+0 (1.8)

where

θi(h) =1− e−bi(n)h

bi(n), i = 1, . . . ,m, n ∈ Z+0 .

It is not difficult to verify that θi(h) > 0 if bi(n), h > 0 and θi(h) ≈ h+ O(h2) for small h > 0. Also, one can show that (1.8)converges towards (1.1) when h→ 0+. In studying the discrete-time analogue (1.8), we assume that

h ∈ (0,+∞), bi : Z→ (0,+∞), τij : Z→ Z+0 , i, j = 1, 2, . . . ,m (1.9)

and the function fi satisfy (H1)–(H4). The system (1.8) is supplemented with initial values given by

xi(s) = ϕi(s), s ∈ Z−0 = 0,−1,−2, . . ., τ = max1≤i,j≤m

supτij(n), n ∈ Z.

In what follows, for convenience, we will use the following notations:

Iω = 0, 1, . . . , ω − 1, u =1ω

ω−1∑k=0

u(k),

where u(k) is an ω-periodic sequence of real numbers defined for k ∈ Z and the notations:

αMij = maxn∈Zαij(n), βMij = maxn∈Z

βij(n), γMij = maxn∈Zγij(n),

M = supu∈R+×Rm×Rm×Rm

|fi(u)|, i = 1, 2, . . . ,m ,

bi = minn∈Iωbi(n), i = 1, 2, . . . ,m, JM = max

n∈Iω|Ii(n)|, i = 1, 2, . . . ,m.

2. Existence of periodic solutions

In this section, based on the Mawhin’s continuation theorem (see [13–17]), we shall study the existence of at least oneperiodic solution of (1.8).Let X , Z be normed vector spaces, L : Dom L ⊂ X → Z be a linear mapping, and N : X → Z be a continuous mapping.

This mapping Lwill be called a Fredholmmapping of index zero if dimKer L = codim Im L <∞ and Im L is closed in Z . If L isa Fredholmmapping of index zero and there exist continuous projectors P : X → X and Q : Z → Z such that Im P = Ker L,KerQ = Im L = Im(I − Q ), it follows that L|Dom L ∩ Ker P : (I − P)X → Im L is invertible, we denote the inverse of thatmap by kP . If Ω is an open bounded subset of X , the mapping N will be called L-compact on Ω if QN(Ω) is bounded andkP(I − Q )N : Ω → X is compact. Since ImQ is isomorphic to Ker L, there exist isomorphisms J : ImQ → Ker L.

Lemma 2.1. Let L be a Fredholm mapping of index zero and let N be L-compact onΩ . Suppose

(a) For each λ ∈ (0, 1) every solution x of Lx = λNx is such that x 6∈ ∂Ω;(b) QNx 6= 0 for each x ∈ ∂Ω ∩ Ker L and degJQN,Ω ∩ Ker L, 0 6= 0.

Then the equation Lx = Nx has at least one solution lying in Dom L ∩Ω .

Theorem 2.1. Assume that (H1)–(H4) and (1.9) hold. Then system (1.8) has at least one ω-periodic solution.


Proof. Define

lm = x = x(k) : x(k) ∈ Rm, k ∈ Z.

|x| =

(m∑i=1

x2i

) 12

, ∀x ∈ Rm.

Let lω ⊂ lm denotes the subspace of allω periodic sequences equipped with the norm ‖ · ‖, i.e., ‖x‖ = maxk∈Iω |x(k)|, for anyx(k) = (x1(k), x2(k), . . . , xm(k))T, k ∈ Z ∈ lω .It is easy to prove that lω is a finite dimensional Banach space.Let

lω0 =

y = y(k) ∈ lω :

1ω

ω−1∑k=0

y(k) = 0

,

lωc =y = y(k) ∈ lω : y(k) = c ∈ Rn, k ∈ Z

,

then it follows that lω0 and lωc are both closed linear subspaces of l

ω and

lω = lω0 ⊕ lωc , dim lωc = m.

Now let us define X = Y = lω, (Lx)(k) = x(k+ 1)− x(k), x ∈ X, k ∈ Z, and

(Nxi)(n) = xi(n)(e−b1(n)h − 1)+ θi(h)

(fi


+∞∑p=1

Hi1(p)x1(n− p), . . . ,+∞∑p=1

Him(p)xm(n− p)

)+ Ii(n)

), i = 1, . . . ,m, n ∈ Z+0 . (2.1)

It is trivial to see that L is a bounded linear operator and

Ker L = lωc , Im L = lω0 ,

as well as

dimKer L = m = codim Im L;

then it follows that L is a Fredholm mapping of index zero.Define

Px =1ω

ω−1∑k=0

x(k), x ∈ X, Qy =1ω

ω−1∑k=0

y(k), y ∈ Y .

It is not difficult to show that P and Q are continuous projectors such that

Im P = Ker L and KerQ = Im L = Im(I − Q ).

Furthermore, the generalized inverse (to L) kP : Im L→ Ker P ∩ Dom L is given by

(kPy)(n) =k−1∑ν=0

y(ν)−1ω

ω∑ν=1

ν−1∑s=0

y(s). (2.2)

Clearly, QN and kP(I − Q )N are continuous, since X is a finite-dimensional Banach space. Using the Arzela–Ascoli Theorem,it is not difficult to show that kP(I − Q )N(Ω) is compact for any open bounded setΩ ⊂ X . Moreover, QN(Ω) is bounded.Thus N is L-compact onΩ . Since ImQ = Ker L, the isomorphic mapping J from ImQ to Ker L is I . We now are in position tosearch for an appropriate open, bounded subsetΩ ⊂ X for the continuation theorem.Corresponding to operator equation Lx = λNx, λ ∈ (0, 1), we have

xi(n+ 1)− xi(n) = λ

(xi(n)(e−bi(n)h − 1)+ θi(h)fi


+∞∑p=1

Hi1(p)x1(n− p), . . . ,+∞∑p=1

Him(p)xm(n− p)

)+ θi(h)Ii(n)

). (2.3)


Suppose that (x1(n), x2(n), . . . , xm(n))T ∈ X is a solution of system (2.3) for a certain λ ∈ (0, 1). In view of (2.3), we have

maxn∈Iω|xi(n)| = max

n∈Iω|xi(n+ 1)|

≤ maxn∈Iω

((1+ λ(e−bih − 1)

)|xi(n)| + λθi(h)

∣∣∣∣∣fi(n, x1(n), . . . , xm(n); x1(n− τi1(n)), . . . , xm(n− τim(n));

+∞∑p=1

Hi1(p)x1(n− p), . . . ,+∞∑p=1

Him(p)xm(n− p)

)∣∣∣∣∣+ λθi(h)|Ii(n)|). (2.4)

Hence

(1− e−bih)maxn∈Iω|xi(n)| ≤ θi(h)[M + JM ], i = 1, 2, . . . ,m,

this is,

maxn∈Iω|xi(n)| ≤

θi(h)[M + JM ]1− e−bih

:= Ai, i = 1, 2, . . . ,m.

Denote A = (∑mi=1 A

2i )12 +B, where B > 0 is a constant. Clearly, A is independent of λ. Nowwe takeΩ = x ∈ X : ‖x‖ < A.

This Ω satisfies condition (i) in Lemma 2.1. When x ∈ ∂Ω ∩ Ker L = ∂Ω ∩ Rm, x is a constant vector in Rm with ‖x‖ = A.Then, if necessary, we can let A be greater such that

xTQNx =m∑i=1

−x2i

ω−1∑s=0

(1− e−bi(s)h)ω

+ xi[fi(t, x1 . . . , xm)+ I i]θi(h)

≤

m∑i=1

[−(1− e−bih)|xi|2 + |xi|(M + JM)θi(h)]

≤ − min1≤i≤m1− e−bih‖x‖2 +

√m(M + JM)θi(h)‖x‖ < 0.

Therefore, QNx 6= 0 for any x ∈ ∂Ω ∩ ker L. Let ψ(γ ; x) = −γ x + (1 − γ )QNx, γ ∈ [0, 1], then for any x ∈ ∂Ω ∩ ker L,xTψ(γ ; x) < 0. From the homotopy invariance of Brouwer degree, it follows that

degJQN,Ω ∩ Ker L, 0 = deg−x,Ω ∩ Ker L, 0 6= 0.

Condition (b) of Lemma 2.1 is also satisfied. Thus, by Lemma 2.1 we conclude that Lx = Nx has at least one solution in X ,that is, (1.8) has at least one ω-periodic solution. The proof is complete.

3. Stability of periodic solutions

In this section, we shall construct appropriate Lyapunov functions to study the stability of periodic solutions of (1.8).

Theorem 3.1. Suppose the assumptions (H1)–(H4) and (1.9) hold, furthermore, assume that τij(n) ≡ τij ∈ Z+, n ∈ Z, i, j =1, 2, . . . ,m, are constants and

bi(n) >

(m∑j=1

αMji +

m∑j=1

βMji +

m∑j=1

γMji

),

+∞∑p=1

pλpHji(p) < +∞,

then the ω-periodic solution x∗(n) = (x∗1(n), x∗

2(n), . . . , x∗m(n))

T of (1.8) is unique and globally exponentially stable in the

sense that there exist constants λ > 1 and δ ≥ 1 such that

m∑i=1

|xi(n)− x∗i (n)| ≤ δ(1λ

)n m∑i=1

sups∈Z−0

|xi(s)− x∗i (s)|

, n ∈ Z+0 . (3.1)


Proof. Let x(n) = (x1(n), x2(n), . . . , xm(n))T be an arbitrary solution of (1.8), and x∗(n) = (x∗1(n), x∗

2(n), . . . , x∗m(n))

T

also be an ω-periodic solution of (1.8). Then

|xi(n+ 1)− x∗i (n+ 1)| ≤ |xi(n)− x∗

i (n)|e−bih + θi(h)

m∑j=1

αMij |xj(n)− x∗

j (n)|

+ θi(h)m∑j=1

βMij |xj(n− τij)− x∗

j (n− τij)|

+ θi(h)m∑j=1

γMij

+∞∑p=1

Hij(p)|xj(n− p)− x∗j (n− p)| i = 1, 2, . . . ,m. (3.2)

Now we consider functions βi(·, ·), i = 1, 2, . . . ,m, defined by

βi(υi, n) = 1− υie−bi(n)h − υiθi(h)m∑j=1

αMji − θi(h)m∑j=1

βMji υτji+1i − θi(h)

m∑j=1

γMji

+∞∑p=1

υp+1i Hji(p),

where υi ∈ [1,+∞), n ∈ Iω, i = 1, 2, . . . ,m. Since

βi(1, n) = 1− e−bi(n)h − θi(h)m∑j=1


βMji − θi(h)m∑j=1

γMji

= θi(h)

(bi(n)−

m∑j=1

αMji −

m∑j=1

βMji −

m∑j=1

γMji

)≥ min1≤i≤mθi(h)η > 0, i = 1, 2, . . . ,m (3.3)

where

η = min1≤i≤m

bi −

m∑j=1

αMji −

m∑j=1

βMji −

m∑j=1

γMji

,

using the continuity of βi(υi, n) on [1,+∞) with respect to υi and the fact that βi(υi, n) → −∞ as υi → +∞ uniformlyin n ∈ Iω, i = 1, 2, . . . ,m, we see that there exists υ∗i (n) ∈ (1,+∞) such that βi(υ

∗

i (n), n) = 0 for n ∈ Iω, i = 1, 2, . . . ,m.By choosing λ = minυ∗i (n), n ∈ Iω, i = 1, 2, . . . ,m, where λ > 1, we obtain βi(λ, n) ≥ 0 for all n ∈ Iω, i = 1, 2, . . . ,mwith the implications

λe−bi(n)h − λθi(h)m∑j=1


βMji λτji+1 − θi(h)

m∑j=1

γMji

+∞∑p=1

λp+1Hji(p) ≤ 1, n ∈ Iω, i = 1, 2, . . . ,m.

Hence

λe−bih − λθi(h)m∑j=1



m∑j=1

γMji

+∞∑p=1

λp+1Hji(p) ≤ 1, i = 1, 2, . . . ,m. (3.4)

Now let us consider

ui(n) = λn|xi(n)− x∗i (n)|

θi(h), n ∈ Z, i = 1, 2, . . . ,m. (3.5)

Using (3.2) and (3.5), we derive that

ui(n+ 1) ≤ λe−bihui(n)+ λm∑j=1

θj(h)αMij uj(n)+m∑j=1

θj(h)βMij λτij+1uj(n− τij)

+

m∑j=1

θj(h)γMij+∞∑p=1

λp+1Hij(p)uj(n− p). (3.6)


We consider the Lyapunov function

V (n) =m∑i=1

(ui(n)+

m∑j=1

θj(h)βMij λτij+1

n−1∑s=n−τij

ui(s)+m∑j=1


Hij(p)λp+1n−1∑s=n−p

ui(s)

). (3.7)

Consider the difference1V (n) = V (n+ 1)− V (n) along (3.7), we obtain

1V (n) =m∑i=1

(ui(n+ 1)− ui(n)+

m∑j=1


(uj(n)− uj(n− τij)

)

+

m∑j=1


Hij(p)λp+1(uj(n)− uj(n− p)

)),

≤

m∑i=1

((λe−bih − 1)ui(n)+ λ

m∑j=1

θj(h)αMij uj(n)

+

m∑j=1

θj(h)βMij λτij+1uj(n)+

m∑j=1


λp+1Hij(p)uj(n)

),

≤ −

m∑i=1

(1− λe−bih − λθi(h)

m∑j=1



m∑j=1

γMji

+∞∑p=1

λp+1Hji(p)

)ui(n),

and by using (3.4) in the above we deduce that1V (n) ≤ 0 for n ∈ Z+0 . From this result and (3.7) it follows thatm∑i=1

ui ≤ V (n) ≤ V (0) for n ∈ Z+.

Thus,m∑i=1

ui = λnm∑i=1

|xi(n)− x∗i (n)|θi(h)

≤

m∑i=1

(ui(0)+

m∑j=1


−1∑s=−τij

ui(s)+m∑j=1


Hij(p)λp+1−1∑s=−p

ui(s)

),

≤

m∑i=1

(1+ θi(h)

m∑j=1

βMji λτji+1τji + θi(h)

m∑j=1

γMji

+∞∑p=1

Hji(p)λp+1p

)sups∈Z−0

|xi(s)− x∗i (s)|.

Therefore, we obtain the assertion (3.1), where

δ =

max1≤i≤mθi(h)

min1≤i≤mθi(h)

σ ≥ 1,

σ = max1≤i≤m

1+ θi(h)

m∑j=1

βMji λτji+1τji + θi(h)

m∑j=1

γMji

+∞∑p=1

Hji(p)λp+1p

≥ 1.

We conclude from (3.1) that the unique periodic solution of (1.8) is globally exponentially stable and this completes theproof.

4. Numerical simulations

In this section, we consider the following discrete-time neural networks with delays:


Example 4.1.

x(n+ 1) = x(n)e−5h[cos(7nπ)+4] +1− e−5h[cos(7nπ)+4]

5[cos(7nπ)+ 4]

y(n)

2+ [y(n)]2+

z(n)3+ [z(n)]2

+x(n− 8)

3+ [x(n− 8)]2

+

100∑p=1(eh − 1)e−phx(n− p)

1+

[100∑p=1(eh − 1)e−phx(n− p)

]2 + 8 sin(nπ),

y(n+ 1) = y(n)e−4h[sin(5nπ)+7] +1− e−4h[sin(5nπ)+7]

4[sin(5nπ)+ 7]

x(n)

1+ [x(n)]2+

y(n)3+ [y(n)]2

+y(n− 1)

1+ [y(n− 1)]2

+

100∑p=1(eh − 1)e−phx(n− p)

2+

[100∑p=1(eh − 1)e−phx(n− p)

]2 + 7 cos(nπ),

z(n+ 1) = z(n)e−6h[sin(3nπ)+6] +1− e−6h[sin(3nπ)+6]

6[sin(3nπ)+ 6]

x(n)h

1+ [x(n)]2+

z(n)h3+ [z(n)]2

+z(n− 6)h

1+ [z(n− 6)]2

+

100∑p=1(eh − 1)e−phx(n− p)h

3+

[100∑p=1(eh − 1)e−phx(n− p)

]2 + 2 sin(nπ),

(4.1)

with initial conditions

(1) ϕ1(n) = 3 sin(πnh)+ 7, ϕ2(n) = 1.8 cos(3πnh)+ 1, ϕ3(n) = sin(2πnh)− 20,(2) ϕ1(n) = 5 sin(2πnh)− 6, ϕ2(n) = 0.8 cos(2πnh)+ 15, ϕ3(n) = sin(4πnh)− 6,(3) ϕ1(n) = 3 sin(πnh)+ 17, ϕ2(n) = 3 cos(πnh)− 15, ϕ3(n) = 2 sin(2πnh)+ 2,

respectively.

It is easy to verify that

bi(n) >

(m∑j=1

αMji +

m∑j=1

βMji +

m∑j=1

γMji

),

and the functionHij(p) satisfy100∑p=1

(eh − 1)e−ph ≈ 1,

and100∑p=1

p(eh − 1)e−ph <+∞∑p=1

p(eh − 1)e−ph < +∞,


so the condition (H4) is satisfied. Furthermore, using the same method as Theorem 3.1, we can choose a constant λ suchthat

100∑p=1

pλpHji(p) =100∑p=1

pλp(eh − 1)e−ph <+∞∑p=1

pλp(eh − 1)e−ph =λ(eh − 1)e−h

(1− λe−h)2

< +∞.

Thus, by using Theorems 2.1 and 3.1, the system (4.1) has a unique positive 2-periodic solution which is globally exponen-tially stable (see Fig. 1).

Example 4.2.

x(n+ 1) = x(n)e−4h[sin(7nπ)+5] +1− e−4h[sin(7nπ)+5]

4[sin(7nπ)+ 5]

x(n)

1+ [x(n)]2+

y(n)2+ [y(n)]2

+z(n)

3+ [z(n)]2

+x(n− 1)

1+ [x(n− 1)]2+

200∑p=1(eh − 1)e−phx(n− p)

1+

[100∑p=1(eh − 1)e−phx(n− p)

]2 + 5 sin(nπ),

y(n+ 1) = y(n)e−4h[sin(5nπ)+7] +1− e−4h[sin(5nπ)+7]

4[sin(5nπ)+ 7]

x(n)

1+ [x(n)]2+

y(n)2+ [y(n)]2

+y(n)

3+ [y(n)]2

+x(n− 1)

1+ [x(n− 1)]2+

100∑p=1(eh − 1)e−phx(n− p)

2+

[100∑p=1(eh − 1)e−phx(n− p)

]2 + 6 cos(nπ),

z(n+ 1) = z(n)e−4h[sin(3nπ)+7] +1− e−4h[sin(3nπ)+7]

4[sin(3nπ)+ 7]

x(n)h

1+ [x(n)]2+

z(n)h2+ [y(n)]2

+z(n)h

3+ [z(n)]2

+x(n− 1)h

1+ [x(n− 1)]2+

100∑p=1(eh − 1)e−phx(n− p)h

3+

[100∑p=1(eh − 1)e−phx(n− p)

]2 + 7 sin(nπ),

(4.2)

with initial conditions

(1) ϕ1(n) = 7, ϕ2(n) = 1, ϕ3(n) = −20,(2) ϕ1(n) = −6, ϕ2(n) = 5, ϕ3(n) = −6,(3) ϕ1(n) = 1, ϕ2(n) = −5, ϕ3(n) = 2,

respectively.

It is straightforward to check that all the conditions needed in Theorems 2.1 and 3.1 are satisfied. Therefore, system (4.2)has exactly one 2-periodic solution which is globally exponentially stable (see Fig. 2).


t

x y

t

z–20

–10

0

10

20

–20

–10

0

10

20

–10 0 10 20 30

t–10 0 10 20 30

–30

–20

–10

0

10

–10 0 10 20 30

Fig. 1. ‘‘−·’’ represents the state trajectory of the neural network (4.1) with initial condition (1); ‘‘· · ·’’ represents the state trajectory of the neural network(4.1) with initial condition (2); ‘‘−’’ represents the state trajectory of the neural network (4.1) with initial condition (3). The three state trajectories of theneural network (4.1) are all converge exponentially to a unique 2-periodic solution (h = 0.025).

xy

t

yz

t

zx

xy

z

–10

0

10

20

t

–10

0

10

20

–20

0

20

–20

0

20

–20

0

20

–20

0

20

–20

0

20

–20

0

20

–20

0

20

–20

0

20

–20

0

20–10

0

10

20

Fig. 2. Seen from the three-dimensional spaces, the neural network (4.2) has a unique globally exponentially stable 2-periodic solution (h = 0.025).


Acknowledgements

The authors would like to thank the Editor and the reviewers for their helpful comments and constructive suggestions,which have been very useful for improving the presentation of this paper.

References

[1] Y.Q. Li, L.H. Huang, Exponential convergence behavior of solutions to shunting inhibitory cellular neural networks with delays and time-varyingcoefficients, Math. Comput. Modelling 48 (2008) 499–504.

[2] W. Wu, B.T. Cui, X.Y. Lou, Global exponential stability of Cohen–Grossberg neural networks with distributed delays, Math. Comput. Modelling (2008)868–873.

[3] H.J. Jiang, J.D. Cao, BAM-type Cohen–Grossberg neural networks with time delays, Math. Comput. Modelling 47 (2008) 92–103.[4] S.J. Guo, L.H. Huang, Stability of nonlinear waves in a ring of neurons with delays, J. Differential Equations 236 (2007) 343–374.[5] Y.R. Liu, Z.D. Wang, X.H. Liu, Asymptotic stability for neural networks with mixed time-delays: the discrete-time case, Neural Netw. 22 (2009) 67–74.[6] S.M. Zhong, X.Z. Liu, Exponential stability and periodicity of cellular neural networkswith time delay, Math. Comput. Modelling 45 (2007) 1231–1240.[7] J.H. Park, A new stability analysis of delayed cellular neural networks, Appl. Math. Comput. 181 (2006) 200–205.[8] M. Fan, X.F. Zou, Global asymptotic stability of a class of nonautonomous integro-differential systems and applications, Nonlinear Anal. 57 (2004)111–135.

[9] S. Mohamad, A.G. Naim, Discrete-time analogues of integrodifferential equationsmodelling bidirectional neural networks, J. Comput. Appl. Math. 138(2002) 1–20.

[10] Y.K. Li, Global stability and existence of periodic solutions of discrete delayed cellular neural networks, Phys. Lett. A 333 (2004) 51–61.[11] S. Mohamad, K. Gopalsamy, Neuronal dynamics in time varying environments: continuous and discrete time models, Discrete Contin. Dyn. Syst. 6

(2000) 841–860.[12] P. Liu, X. Cui, A discrete model of competition, Math. Comput. Simulation 49 (1999) 1–12.[13] R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977.[14] T. Yoshizawa, Stability theory by Liapunov’s second method, Math. Soc. Japan, Tokyo, 1966.[15] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost-Periodic Solutions, in: Applied Math. Sciences, vol. 14, Springer-

Verlag, 1975.[16] M. Fan, K. Wang, Periodic solutions of a discrete time nonautonomous ratio-dependent predator–prey system, Math. Comput. Modelling 35 (2002)

951–961.[17] M. Fan, K.Wang, Existence and global attractivity of positive periodic solutions of periodic n-species Lotka–Volterra competition systemswith several

deviating arguments, Math. Biosci. 160 (1999) 47–61.

numerical simulation of periodic solutions for a class of numerical discretization neural networks

Documents