Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales

https://doi.org/10.1016/j.cnsns.2010.11.004Get rights and content

Abstract

By using the method of coincidence degree and constructing suitable Lyapunov functional, several sufficient conditions are established for the existence and global exponential stability of anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scale T. Our results are new even if the time scale T=R or Z. An example is given to illustrate our feasible results.

Research highlights

► We study the existence of ant-periodic solutions to impulsive neural networks on time scales. ► We unify the discrete and continuous time shunting inhibitory neural networks under one frame work. ► However, most of the known work deals with continuous time neural networks without impulse. ► Our results are new even if the time scale T=R or Z.

Introduction

Since Bouzerdout and Pinter in [1] described SICNNs as a new cellular neural networks (CNNs), SICNNs have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Hence, they have been the object of intensive analysis by numerous authors in recent years. Many important results on the dynamical behaviors of SICNNs have been established and successfully applied to signal processing, pattern recognition, associative memories, and so on. We refer the reader to [2], [3], [4], [5], [6], [7], [8], [9] and the references cited therein.

In contrast, however, very few results are available on the existence and exponential stability of anti-periodic solutions for neural networks, while the existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [10], [11], [12], [13], [14]). Since SICNNs can be analog voltage transmission, and voltage transmission process often a anti-periodic process. Thus, it is worth while to continue to investigate the existence and stability of anti-periodic solutions of SICNNs.

In reality, many physical systems undergo abrupt changes at certain moments due to instantaneous perturbations, which lead to impulsive effects. Since the existence of impulses is frequently a source of instability, bifurcation and chaos for neural networks, the impulsive neural networks is an appropriate description of the phenomena of abrupt qualitative dynamical changes of essentially continuous-time systems, see [15], [16], [17], [18], [19] and references therein.

As well known, both continuous and discrete systems are very important in implementing and applications. Recently, several types of neural networks on time scales have been presented and studied, see, for e.g. [20], [21], [22], [23], which can unify the continuous and discrete situations.

Motivated by all above mentioned, in this paper, we will apply the method of coincidence degree to investigate the existence of anti-periodic solutions to the following impulsive shunting inhibitory cellular neural networks on time scalesxijΔ(t)=-aij(t)xij(t)-CklNr(i,j)Cijkl(t)0+Kij(u)xij(t)fij(xkl(t-u))Δu+Lij(t),tT+,tth,hN,Δxij(th)=xij(th+)-xij(th-)=Iijk(xij(th)),t=th,i=1,,m,j=1,,n,where T is an ω2-periodic time scale which has the subspace topology inherited from the standard topology on R and T+={tT:t0},Cij denotes the cell at the (i, j) position of the lattice, the r-neighborhood Nr(i, j) of Cij is given byNr(i,j)={Cij:max(|k-i|,|l-j|)r,1km,1ln},xij acts as the activity of the cell Cij, Lij(t) is the external input to Cij, aij(t) > 0 represents the passive decay rate of the cell activity, Cijkl(t)>0 is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cij, and the activity function fij( · ) is a continuous function representing the output or firing rate of the cell Ckl,xij(th+),xij(th-) represent the right and left limit of xij(th) in the sense of time scales, {th} is a sequence of real numbers such that 0 < t1 < t2 <⋯<  th  ∞ as h  ∞, there exists a positive integer p such that th+p=th+ω2,Iij(h+p)(xij(th+p))=-Iijh(-xij(th)),hN. Without loss of generality, we also assume that [0,ω2)T{th:hN}={t1,t2,,tq}.

The main purpose of this paper is to study the existence and global exponential stability of the anti-periodic solutions of system (1.1) by using the method of coincidence degree and Lyapunov method. To the best of the author’s knowledge, this is the first paper to discuss the anti-periodic solutions of shunting inhibitory cellular neural networks with impulse on time scales.

Let x(t)=(x11(t),x12(t),,x1n(t),,xm1(t),xm2(t),,xmn(t))TC(T,Rmn), we define the norm x=i=1mj=1nmaxt[0,ω]T|xij(t)|. The initial conditions associated with system (1.1) are of the formxij(t)=φij(t),t[-,0]T,where φij(t), i = 1, 2,  , m, j = 1, 2,  , n are continuous functions on [-,0]T.

For the sake of convenience, we introduce some notationsa̲ij=mint[0,ω]T|aij(t)|,a¯ij=maxt[0,ω]T|aij(t)|,L¯ij=maxt[0,ω]T|Lij(t)|,C¯ijkl=maxt[0,ω]T|Cijkl(t)|,Eij=a̲ijω(1-a¯ijω)-(a̲ijω+1)CklNr(i,j)C¯ijklMfij0+ω|Kij(u)|Δu+h=12qρijh,Dij=(1+a̲ijω)h=12q|Iijh(0)|+ωL¯ij,g2=0ω|g(t)|2Δt1/2,where i = 1, 2,  , m, j = 1, 2,  , n, g is an ω-periodic function.

Throughout this paper, we assume that

  • (H1)

    aijC(T,(0,+)),aij(t+ω2)=aij(t),Cijkl,LijC(T,R),Cijkl(t+ω2)=-Cijkl(t),Lij(t+ω2)=-Lij(t),i=1,2,,m,j=1,2,,n;

  • (H2)

    0+|Kij(u)|Δu<+,i=1,2,,m,j=1,2,,n;

  • (H3)

    fijC(R,R),fij(-u)=fij(u), and there exist positive constants Mfij and Lfij such that |fij(u)|Mfij,|fij(u)-fij(v)|Lfij|u-v|,i=1,2,,m,j=1,2,,n;

  • (H4)

    IijhC(R,R) and there exist positive constants ρijh such that|Iijh(u)-Iijh(v)|ρijh|u-v|

    for all u,vR,hN,i=1,2,,m,j=1,2,,n.

The organization of this paper is as follows. In Section 2, we introduce some definitions and lemmas. In Section 3, by using the method of coincidence degree, we obtain the existence of the anti-periodic solutions of system (1.1). In Section 4, we give the criteria of global exponential stability of the anti-periodic solutions of system (1.1). In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections 3 Existence of anti-periodic solutions, 4 Global exponential stability of the anti-periodic solution. The conclusions are drawn in Section 6.

Section snippets

Preliminaries

In this section, we shall first recall some basic definitions, lemmas which are used in what follows.

Definition 2.1

[24]

A time scale T is an arbitrary nonempty closed subset of the real set R with the topology and ordering inherited from R. The forward and backward jump operators σ,ρ:TT and the graininess μ:TR+ are defined, respectively, byσ(t)inf{sT:s>t},ρ(t)sup{sT:s<t},μ(t)σ(t)-t.The point tT is called left-dense, left-scattered, right-dense or right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t or σ(t) > t,

Existence of anti-periodic solutions

Theorem 3.1

Assume that (H1)–(H4) hold. Suppose further that Eij > 0, i = 1, 2,  , m, j = 1, 2,  , n. Then system (1.1) has at least one ω2-anti-periodic solution.

Proof

Let Ck([0,ω;t1,t2,,tq,tq+1,,t2q]T,Rmn)={x:[0,ω]TRmn|x(k)(t) is a piecewise continuous map with first-class discontinuous points in [0,ω]T{th:hN} and at each discontinuous point it is continuous on the left}, k = 0, 1. TakeX=xC([0,ω;t1,t2,,tq,tq+1,,t2q]T,Rmn):x(t+ω2)=-x(t)for allt[0,ω2]TandY=X×R(mn)×qbe two Banach spaces equipped with the normsxX=i=1

Global exponential stability of the anti-periodic solution

In this section, we will construct some suitable Lyapunov functions to study the global exponential stability of the anti-periodic solution of system (1.1).

Theorem 4.1

Assume that (H1)–(H6) hold. Suppose further that

  • (H7)

    The impulsive operators Iik(xi(t)) satisfyIik(xi(tk))=-γikxi(tk),0γik2,i=1,2,,n,kN.

  • (H8)

    There exist constants ϵ > 0 and η > 0 such that,βij=ϵ-a̲ij+CklNr(i,j)C¯ijklMfij0+|Kij(u)|Δu+Lfij0+|Kij(u)|eϵ(u,α)ΔuMij1+ω2ϵ<-η<0,where Mij=DijEij,i=1,2,,m,j=1,2,,n.

Then the ω2-anti-periodic solution of

An example

In this section, we give an example to illustrate that our results are feasible.

When T=R, consider the following SICNNs with impulsesdxij(t)dt=-aij(t)xij(t)-CklN1(i,j)Cijkl(t)0+Kij(u)xij(t)fij(xkl(t-u))Δu+Lij(t),Δxij(th)=xij(th+)-xij(th-)=-0.025xij(th),i=1,2,j=1,2,h=1,2,where(aij)2×2=2.0+0.02|sin(8πt)|2.0-0.01|cos(8πt)|1.9+0.01|sin(8πt)|1.95-0.01|cos(8πt)|,(Cij)2×2=0.6sin(8πt)0.9sin(8πt)0.8cos(8πt)-0.5cos(8πt),f(u)=12|sin(u)|,(Lij)2×2=0.6sin(8πt)0.5cos(8πt)0.7cos(8πt)0.4sin(8πt),(Kij)2×2=0.2

Conclusions

Arising from problems in applied sciences, it is well-known that the existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations as a special periodic solution and have been extensively studied by many authors during the past ten years, see [10], [11], [12], [13], [14] and references therein. For example, anti-periodic trigonometric polynomials are important in the study of interpolation problems [30], and anti-periodic wavelets are

References (31)

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    During hardware implementation of shunting inhibitory cellular neural networks, the signal transmission process is usually described as an anti-periodic process [21–25]. Anti-periodic oscillatory phenomena for SICNNs involving bounded time-varying delays [22,24–28], continuously distributed delays [23,29] have been studied intensively. However, to the best of our knowledge, very few papers consider the problems of anti-periodic solutions for SICNNs involving proportional delays.

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This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.

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