EXISTENCE AND GLOBAL STABILITY OF PERIODIC SOLUTION FOR DELAYED DISCRETE HIGH-ORDER HOPFIELD-TYPE NEURAL NETWORKS

EXISTENCE AND GLOBAL STABILITY OF PERIODIC SOLUTION FOR DELAYED DISCRETE HIGH-ORDER HOPFIELD-TYPE NEURAL NETWORKS

HONG XIANG, KE-MING YAN, AND BAI-YAN WANG Received 6 February 2005

By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high- order Hopfield-type neural networks. We obtain some easily verifiable sufficient condi- tions to ensure that there exists a unique periodic solution, and all theirs solutions con- verge to such a periodic solution.

1. Introduction

It is well known that studies on neural dynamical systems not only involve discussion of stability property, but also involve other dynamics behaviors such as periodic oscillatory, bifurcation and chaos. In many applications, the property of periodic oscillatory solu- tions are of great interest. For example, the human brain has been in periodic oscillatory or chaos state, hence it is of prime importance to study periodic oscillatory and chaos phenomenon of neural networks. Recently, Liu and Liao [8], Zhou and Liu [15] consider the existence and global exponential stability of periodic solutions of delayed Hopfield neural networks and delayed cellular neural networks. Liu et al. [7] address the existence and global exponential stability of periodic solutions of delayed BAM neural networks.

Since high-order neural networks have stronger approximation property, faster conver- gence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks, they have attracted considerable attention (see, e.g., [1,2,4,5,10,11,13,14]).

In our previous paper [12], we study the global exponential stability and existence of periodic solutions of the following high-order Hopfield-type neural networks

dxi(t)

dt ^{= −}a_i(t)x_i(t) + m j=1

b_{i j}(t)f_jx_j(t) +

m j=1

b_{i j}(t)f_jx_jt−τ_{i j}(t)+ m j=1

m l=1

e_{i jl}(t)f_jx_j(t)f_lx_l(t) +

m j=1

m l=1

e_{i jl}(t)f_jx_jt−σ_{i jl}(t)f_lx_lt−σ_{i jl}(t)+I_i(t),

(1.1)

Copyright©2005 Hindawi Publishing Corporation

Discrete Dynamics in Nature and Society 2005:3 (2005) 281–297 DOI:10.1155/DDNS.2005.281

wherei=1, 2,...,m,t >0,x_i(t) denotes the potential (or voltage) of the celliat timet.

a_i(t) are positiveω-periodic functions, they denote the rate with which the cellireset their potential to the resting state when isolated from the other cells and inputs.bi j(t),ei jl(t) are the first-and second-order connection weights of the neural network, respectively;

I_i(t) denote theith component of an external input source introduced from outside the network to the celli.

In [6], Li investigates global stability and existence of periodic solutions of discrete de- layed cellular neural networks. However, few authors have studies the dynamical behav- iors of the discrete-time analogues of delayed high-order Hopfield-type neural networks with variable coefficient. In this paper, we are concerned with the following discrete ana- logue of (1.1) of the form

x_i(n+ 1)=e^−ai(n)hx_i(n) +θ_i(h) n j=1

b_{i j}(n)f_jx_j(n) +θi(h)

n j=1

bi j(n)fj

xj

n−τi j(n)+θi(h) n j=1

n l=1

ei jl(n)fj

xj(n)fl

xl(n)

+θi(h) n j=1

n l=1

ei jl(n)fj

xj

n−σi jl(n)fl

xl

t−σi jl(n) +θi(h)Ii(n), i=1, 2,...,m,

(1.2) in whichθi,ai,bi j,ei jl,i,j,l=1, 2,...,m, will be specified in the next section.

With the help of Mawhin’s continuation theorem of coincidence degree theory and constructing Lyapunov functional, we obtain some sufficient conditions ensure that for the discrete networks (1.2) there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution. To the best of our knowledge, this is the first time to study the existence and global attractivity of the periodic solution for the discrete-time analogues of delayed high-order Hopfield-type neural networks with variable coefficient.

The tree of this paper is as follows. InSection 2, following the semi-discretization tech- nique [6,9], we obtain a discrete-time analogue of (1.1). InSection 3, with the help of Mawhin’s continuation theorem of coincidence degree theory, we study the existence of the periodic solution of (1.2). InSection 4, by constructing Lyapunov functional, we de- rive sufficient conditions to ensure that the periodic solution of (1.2) is globally asymp- totically stable.

2. Discrete-time analogues

There is no unique way of deriving discrete time version of dynamical equations corre- sponding to continuous time formulation. First, following [6,9], we reformulate system (1.1) by an approximation of the form

dxi(t) dt ^{= −}a_i

t h

h

x_i(t) + m j=1

b_{i j} t

h

h

f_j

x_j t

h

h

+ m j=1

b_{i j} t

h

h

f_j

x_j t

h

h

−τ_{i j} t

h

h

+ m j=1

m l=1

e_{i jl} t

h

h

f_j

x_j t

h

h

f_l

x_l t

h

h

+ m j=1

m l=1

ei jl

t h

h

fj

xj

t h

h

−σi jl

t h

h

×fl

xl

t h

h

−σi jl

t h

h

+Ii

t h

h

,

(2.1)

wherehis a positive number denoting a uniform discretization step size and [t/h] denotes the greatest integer int/h. For convenience, we denote [t/h]=n,n∈Z0⁺, andai(nh)= a_i(n),b_{i j}(nh)=b_{i j}(n),b_{i j}(nh)=b_{i j}(n),e_{i jl}(nh)=e_{i jl}(n),e_{i jl}(nh)=e_{i jl}(n),τ_{i j}(nh)=τ_{i j}, σi jl(nh)=σi jl,xi(nh)=xi(n),Ii(nh)=Ii. Thus (2.1) takes on the form

dx_i(t)

dt ^{= −}ai(n)xi(t) + m j=1

bi j(n)fj

xj(n)

+ m j=1

bi j(n)fj

xj

n−τi j(n)

+ m j=1

m l=1

ei jl(n)fj

xj(n)fl xl(n) +

m j=1

m l=1

ei jl(n)fj

xj

n−σi jl(n)

×f_lx_ln−σ_{i jl}(n)+I_i(n), n∈Z0⁺.

(2.2)

Integrate it over the interval [nh,t] fort <(n+ 1)hto obtain

xi(t)e^ai(n)t−xi(n)e^ai(n)nh

=e^ai(n)t−e^ai(n)nh ai(n)

m j=1

bi j(n)fj

xj(n)+ m j=1

bi j(n)fj

xj

n−τi j(n)

+ m j=1

m l=1

ei jl(n)fj

xj(n)fl

xl(n)

+ m j=1

m l=1

ei jl(n)fj

xj(n)fl

xl(n)fl

xl

n−σi jl(n)

+I_i(n)

, i=1, 2,...,m.

(2.3)

We lett→(n+ 1)hand obtain

xi(n+ 1)=xi(n)e^−ai(n)h+θi(h) m j=1

bi j(n)fj

xj(n)

+θi(h) m j=1

bi j(n)fj

xj

n−τi j(n)

+θi(h) m j=1

m l=1

ei jl(n)fj

xj(n)fl

xl(n)

+θi(h) m j=1

m l=1

ei jl(n)fj

xj

n−σi jl(n)fl

xl

n−σi jl(n) +θ_i(h)I_i(n), i=1, 2,...,m,n∈Z0⁺,

(2.4)

where

θ_i(h)=1−e^−ai(n)h

a_i(n) , i=1, 2,...,m,n∈Z0⁺. (2.5) It is not difficult to verify thatθ_i(h)>0 ifa_i,h >0 andθ_i(h)≈h+o(h²) for smallh >0.

Also, one can see that (1.2) converges towards (1.1) whenh→0⁺. The system (1.2) is supplemented with initial values given by

xi(s)=ϕi(s), s∈

−τ^∗, 0_Z,τ^∗= max

1_≤i,j,l≤m

maxn∈Z

τi j(n),σi jl(n). (2.6)

In this paper, we assume that

(H1)ai:Z→(0,∞),bi j,bi j,ei jl,ei jl,Ii∈Z→R,τi j,σi jl:Z→Z0⁺,i,j,l=1, 2,...,m, h∈(0,∞).

(H2) f_jare Lipschitzian with Lipschitz constantsL_j>0,

f_j(x)−f_j(y)≤L_j|x−y| (2.7)

for anyx,y∈R, (j∈ {1,...,m}).

(H3) There exist positive constantsNj>0,j∈ {1,...,m}such that

fj(x)≤Nj, j∈ {1,...,m}. (2.8)

For convenience, we will introduce the notation:

I_ω= {0, 1,...,ω−1}, u= 1 ω

ω−1 k=0

u(k), (2.9)

whereu(k) anω-perodic sequence of real numbers defined fork∈Zand notations:

a_i=min

n∈Iω

ai(n), i=1, 2,...,m, I^M=max

n∈Iω

I_i(n),i=1, 2,...,m, M=sup

u∈R

f_j(u), j=1, 2,...,m, b^M_{i j} =max

n∈Iω

b_{i j}(n), b^M_{i j} =max

n∈Iω

b_{i j}(n) e^M_{i jl}=max

n∈Iω

ei jl(n), e^M_{i jl}=max

n∈Iωei jl(n).

(2.10)

3. Existence of periodic solution

In this section, based on the Mawhin’s continuation theorem and Lyapunov functional, we will study the existence of periodic solutions of discrete-time high-order Hopfield- type neural networks (1.2).

First, we will make some preparations.

LetXandZbe two Banach spaces. Consider an operator equation

Lx=λNx, λ∈(0, 1), (3.1)

whereL: DomL∩X→Zis a linear operator andλis a parameter. LetPandQdenote two projectors such that

P:X∩DomL−→KerL, Q:Z−→ Z

ImL. (3.2)

Denote byH: ImQ→KerLis an isomorphism of ImQonto KerL. In the sequel, we will use the following result of Mawhin [3, page 40].

Lemma3.1. LetX andZbe two Banach spaces andLa Fredholm mapping of index zero.

Assume thatN:Ω→ZisL-compact onΩwithΩopen bounded inX. Furthermore assume:

(a)for eachλ∈(0, 1),y∈∂Ω∩DomL,

Lx=λNx; (3.3)

(b)for eachx∈∂Ω∩KerL,

QNx=0, deg{HQNx,Ω∩KerL, 0} =0. (3.4) Then the equationLx=Nxhas at least one solution indomL∩Ω.

Recall that a linear mapping L: DomL∩x→Z with KerL=L⁻¹(0) and ImL= L(DomL), will be called a Fredholm mapping if the following two conditions hold:

(i) KerLhas a finite dimension;

(ii) ImLis closed and has a finite codimension.

Recall also that the codimension of ImLis the dimension ofZ/ImL, that is, the di- mension of the cokernel cokerLofL.

WhenLis a Fredholm mapping, its index is the integer IndL=dim kerL-codim ImL.

We will say that a mappingNisL-compact onΩif the mappingQN:Ω→Zis contin- uous,QN(Ω) is bounded, andK_p(I−Q)N:Ω→Y is compact, that is, it is continuous andK_p(I−Q)N(Ω) is relatively compact, whereK_p: ImL→DomL∩KerPis a inverse of the restrictionLpofLto DomL∩KerP, so thatLKp=IandKpL=I−P.

Next, we will state and prove the existence of periodic solutions of system (1.2).

Theorem3.2. Assume that the condition(H1),(H2), and(H3)are satisfied. Furthermore, assume that

(H4)a_i,b_{i j},b_{i j},e_{i jl},e_{i jl},I_i,i,j,l=1, 2,...,m, are allω-periodic functions.

Then system (1.2) has at least oneω-periodic solution.

Proof. Similar to that of [6], we define

lm= x=

x(k):x(k)∈R^m,k∈Z. (3.5)

Letl^ω⊂l_mdenote the subspace of allωperiodic sequences equipped with the usual supre- mum norm · , that is,

x =

x1(k),...,xm(k)^T

=^m

i=1

maxk∈Iω

xi(k), for anyx=

x1(k),...,xm(k):k∈Z∈l^ω.

(3.6)

It is not difficult to show thatl^ωis a finite-dimensional Banach space.

Let

l^ω0 = x=

x(k)∈l^ω:

ω−1 k=0

x(k)=0

, l^ω_c =

x=

x(k)∈l^ω:x(k)=h∈R^m,k∈Z,

(3.7)

then it follows thatl0^ωandl^ω_c are both closed linear subspaces ofl^ωand

l^ω=l0^ω⊕l^ω_c, diml^ω_c =m. (3.8)

In order to useLemma 3.1to system (1.2), we takeX=Y=l^ω, (Lx)(k)=x(k+ 1)−x(k), and let

Nx(n)=













x1(n)e^−a1(n)h−1+θ1(h) m j=1

b1j(n)f_jx_j(n)+ ...

x_m(n)e^−am(n)h−1+θ_m(h) m j=1

b_{m j}(n)f_jx_j(n)+ θ1(h)

m j=1

b1j(n)f_jx_jn−τ1j(n)+θ1(h)I1(n) ...

θ_m(h) m j=1

b_{m j}(n)f_jx_jn−τ_{m j}(n)+θ_m(h)I_m(n) +θ1(h)

m j=1

m l=1

e1jl(n)f_jx_j(n)f_lx_l(n)+ ...

+θ_m(h) m j=1

m l=1

e_{m jl}(n)f_jx_j(n)f_lx_l(n)+ θ1(h)

n j=1

n l=1

e1jl(n)f_jx_jn−σ1jl(n)f_lx_lt−σ1jl(n) ...

θ_m(h) n j=1

n l=1

e_{m jl}(n)f_jx_jn−σ_{m jl}(n)f_lx_lt−σ_{m jl}(n)











 .

(3.9)

It is trivial to see thatLis a bounded linear operator and

KerL=l^ω_c, ImL=l0^ω, (3.10)

as well as

dim KerL=m=codim ImL; (3.11)

then it follows thatLis a Fredholm mapping of index zero.

Define

Px= 1 ω

ω−1 s=0

x(s), x∈X, Qz= 1 ω

ω−1 s=0

z(s), z∈Y. (3.12) It is not difficult to show thatPandQare continuous projectors such that

ImP=KerL, ImL=KerQ=Im(I−Q). (3.13)

Furthermore, the generalized inverse (toL)K_p: ImL→KerP∩domLhas the form

K_p(z)=^k− 1 s=0

z(s)− 1 ω

ω−1 s=0

(ω−s)z(s). (3.14)

Clearly, QN and K_p(I−Q)N are continuous. Since X is a finite-dimensional Banach space, using the Arzela-Ascoli theorem, it is not difficult to show thatQN(Ω),K_p(I− Q)N(Ω) are relatively compact for any open bounded setΩ⊂X. Hence,NisL-compact onΩ, hereΩis any open bounded set inX.

Now we reach the position to search for an appropriate open, bounded subsetΩfor the application of the Lemma 3.1. Corresponding to the operator equationLx=λNx, λ∈(0, 1), we have

xi(n+ 1)−xi(n)=λ

xi(n)e^−ai(n)h−1+θi(h) m j=1

bi j(n)fj

xj(n)

+θi(h) m j=1

bi j(n)fj

xj

n−τi j(n)

+θi(h) m j=1

m l=1

ei jl(n)fj

xj(n)fl

xl(n)

+θi(h) m j=1

m l=1

ei jl(n)fj

xj

n−σi jl(n)fl

xl

n−σi jl(n)

+θ_i(h)I_i(n)

, i=1, 2,...,m.

(3.15)

Assume that x(n)=(x1(n),...,x_m(n))∈X is a solution of system (3.15) for some λ∈ (0, 1), from (3.15), we obtain

maxn∈Iω

x_i(n)=max

n∈Iω

x_i(n+ 1)

≤

1 +λe^−ai(n)h−1xi(n)+λθi(h) m j=1

bi j(n)fj

xj(n

+λθi(h) m j=1

bi j(n)fj

xj

n−τi j(n)

+λθi(h) m j=1

m l=1

ei jl(n)fj

xj(n)fl

xl(n)

+λθ_i(h) m j=1

m l=1

e_{i jl}(n)f_jx_jn−σ_{i jl}(n)f_lx_ln−σ_{i jl}(n)+λθ_i(h)I_i(n)

≤

1 +λe^−ai(n)h−1max

n∈Iω

xi(n)+λθi(h)mbM+λθi(h)mbM +λθ_i(h)m²eM²+λθ_i(h)m²eM²+λθ_i(h)I^M, i=1, 2,...,m.

(3.16) Hence,

1−e^−ai(n)hmax

n∈Iω

xi(n)

≤θi(h)mbM+mbM+m²eM²+m²eM ²+I^M, i=1, 2,...,m.

(3.17) That is

maxn∈Iω

xi(n)≤θi(h)mbM+mbM +m²eM²+m²eM²+I^M

1−e^−ai(n)h :=Ai. (3.18) DenoteA=_m

i=1A_i+E, whereEis taken sufficiently large such thatx< A, clearly,A is independent ofλ. Now, we takeΩ= {u∈X:x< A}. It is clear thatΩsatisfies the requirement (a) inLemma 3.1.

Whenx∈∂Ω∩KerL,x is a constant vector inR^m withx =A. Furthermore, take H: ImQ→KerL,r→r. we can letAbe greater such that

x1,...,x_mHQN





 x1

... x_m







=ⁿ

i=1

−x_i²

ω−1 s=0

e^−ai(s)h−1 ω

+θi(h)xi

m j=1

bi jfj xj

xi

+θi(h)xi

m j=1

bi jfj xj

xi+θi(h) m j=1

m l=1

ei jlfj xj

fl xl

xi

+θi(h) m j=1

m l=1

ei jlfj

xj

fl

xl

xi+θi(h)Iixi

<0.

(3.19)

So for anyx∈∂Ω∩KerL,QNx=0. Furthermore, letΨ(r;u)= −rx+ (1−r)JQNx, then for anyx∈∂Ω∩KerL,x^TΨ(r;x)<0, we get

deg{HQNx,Ω∩KerL, 0} =deg{−x,Ω∩KerL, 0} =0. (3.20) Condition (b) ofLemma 3.1is also satisfied. By now we have prove that Ωsatisfies all the requirements inLemma 3.1. Hence, system (1.2) has at least oneω-periodic solution.

The proof is complete.

4. Global stability of the periodic solution

In this section, we will obtain sufficient conditions for the global asymptotic stability and global exponential stability of the periodic solution of discrete high-order Hopfield-type networks (1.2).

Theorem4.1. Assume that condition(H1),(H2),(H3), and(H4)are satisfied. Further- more, assume thatτ_{i j}(n)=τ_{i j}∈Z⁺,σ_{i jl}(n)=σ_{i jl}∈Z⁺, and

(H5)There exists a positive real number sequenceαisuch that λi=αi

1−e^ai−Li

m j=1

αjθj(h)b^M_ji −Li

m j=1

αjθj(h)b^M_ji −Li

m j=1

m l=1

αjθj(h)e^M_jilM

−Li

m j=1

m l=1

αjθj(h)e^M_jilM−Li

m j=1

m l=1

αjθj(h)e^M_jliM

−Li

m j=1

m l=1

αjθj(h)e^M_jliM >0, i=1, 2,...,m.

(4.1)

Then theω-periodic solution of (1.2) is unique and all other solutions of (1.2) converges to its uniqueω-periodic solutions.

Proof. According toTheorem 3.2, we know that (1.2) has aω-periodic solutionx^∗(n)= (x^∗1(n),x^∗2(n),...,x^∗_m(n))^T. Obviously, if this periodic solution is globally attractivity, then it is unique. Letx(n)=(x1(n),x2(n),...,xm(n))^T is an arbitrary solution of (1.2) and let

xi(n+ 1)−x_i^∗(n+ 1)= −e^ai(n)hxi(n)−x_i^∗(n) +θi(h)

m j=1

bi j(n)fj

xj(n)−fj

x^∗_j(n)

+θi(h) m j=1

bi j(n)fj

xj

n−τi j

−fj

x^∗_jn−τi j

+θi(h) m j=1

m l=1

ei jl(n)fj

xj(n)fl

xl(n)−fj

x^∗_j(n)fl

x^∗_l (n)

+θi(h) m j=1

m l=1

ei jl(n)fj

xj

n−σi jl

fl

xl

n−σi jl

+f_jx^∗_jn−σ_{i jl}f_lx^∗_ln−σ_{i jl}, i=1, 2,...,m.

(4.2) Hence,

xi(n+ 1)−x^∗_i (n+ 1)

≤ −e^aixi(n)−x_i^∗(n) +θ_i(h)

m j=1

b^M_{i j}L_jx_j(n)−x^∗_j(n) +θ_i(h)

m j=1

b^M_{i j}L_jx_jn−τ_{i j}−x^∗_jn−τ_{i j}