# Oscillatory Behavior on a Three-Node Neural Network Model with Discrete and Distributed Delays

Authors: Chunhua Feng
Journal: Advances in Artificial Neural Systems
 (2014)

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## Abstract

This paper investigates the oscillatory behavior of the solutions for a three-node neural network with discrete and distributed delays. Two theorems are provided to determine the conditions for oscillating solutions of the model. The criteria for selecting the parameters in this network are derived. Some simulation examples are presented to illustrate the effectiveness of the results.


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## Body

## 1. Introduction

Neural networks are complex and large-scale dynamical systems. Time delay is inevitably encountered in implementation of dynamical neural networks and is frequently a source of oscillation and instability. On the one hand, information transmission from one neuron to another neuron may make the response of networks with discrete delays. On the other hand, neural networks usually have spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. Thus, the distributed delays in neural system need to be considered. In the past two decades, many researches have studied various neural networks with discrete and distributed delays [1–4]. In [1], Gopalsamy and Leung studied the following neural network model:
(1)dx1(t)dt=-x1(t)+atanh⁡[x2(t)-bx2(t-τ)-c],dx2(t)dt=-x2(t)+atanh⁡[x1(t)-b∫0∞F(s)x1(t-s)ds-c].By means of the Lyapunov functional, the authors obtained some necessary and sufficient conditions for the existence of a globally asymptotically stable equilibrium point of system (1). Ruan and Filfil [2] considered a two-neuron network model with multiple discrete and distributed delays as follows:
(2)1a10dx1(t)dt+x1(t)=F{∫-∞tx1(τ)k11(t-τ)dτf1+c12x2(t-σ12)+b11∫-∞tx1(τ)k11(t-τ)dτ},1a20dx2(t)dt+x2(t)=F{∫-∞tx2(τ)k22(t-τ)dτf2+c21x1(t-σ21)+b22∫-∞tx2(τ)k22(t-τ)dτ},
where the delayed feedback kernel satisfies ∫0∞kii(s)ds=const. and ∫0∞skii(s)ds<∞,F(u)=(1/(1+e-u))(i=1,2). Local stability analyses are carried out for model (2). Computer simulations are performed to illustrate the obtained results. Liao et al. [3] discussed a two-neuron system with distributed delays in the frequency domain as follows:
(3)dx1(t)dt=-x1(t)+a1f[x2(t)-b2∫0∞F(r)x2(t-r)dr-c1],dx2(t)dt=-x2(t)+a2f[x1(t)-b1∫0∞F(r)x1(t-r)dr-c2],
where F(r) is a strong kernel. For simplicity, the authors set c1=c2=0. Let
(4)y1(t)=x1(t)-b1∫0∞F(r)x1(t-r)dr,y2(t)=x2(t)-b2∫0∞F(r)x2(t-r)dr.Then system (3) is equivalent to the following model:
(5)dy1(t)dt=-y1(t)+a1f[y2(t)]-a1b2∫0∞F(r)f[y2(t-r)]dr,dy2(t)dt=-y2(t)+a2f[y1(t)]-a2b1∫0∞F(r)f[y2(t-r)]dr.By applying the frequency domain approach and analyzing the characteristic equation, the Hopf bifurcation occurs when the mean delayμ exceeds a critical value. Thus, there is a family of periodic solutions bifurcates from the equilibrium point. Recently, Hajihosseini et al. [4] have investigated the following three-node network model:
(6)dx1(t)dt=-x1(t)+tanh[∫0∞F(r)x2(t-r)dr],dx2(t)dt=-x2(t)+tanh[∫0∞F(r)x3(t-r)dr],dx3(t)dt=-x3(t)+w1tanh[∫0∞F(r)x1(t-r)dr]+w2tanh[∫0∞F(r)x2(t-r)dr],
where w1 and w2 are parameters. The authors also make the change of variables yi(t)=∫0∞F(r)xi(t-r)dr(i=1,2,3) such that system (6) changes to the following:
(7)dy1(t)dt=-y1(t)+∫0∞F(r)tanh⁡[y2(t-r)]dr,dy2(t)dt=-y2(t)+∫0∞F(r)tanh⁡[y3(t-r)]dr,dy3(t)dt=-y3(t)+w1∫0∞F(r)tanh⁡[y1(t-r)]dr+w2∫0∞F(r)tanh⁡[y2(t-r)]dr.Under the restrictive condition|w1+w2|<1, the Hopf bifurcation and stability of the bifurcating periodic solutions have been discussed. There are many authors who have studied the bifurcating periodic solutions for various models [5–14]. It is well known that the basic method in studying the bifurcating periodic solution for a time delay system is to discuss an algebraic equation in which the bifurcating value will be determined. Motivated by the above models we shall concern the existence of oscillating solutions for the following three-node network system with discrete and distributed delays:
(8)dx1(t)dt=-a1x1(t)+b2f[x2(t-τ2)]+c3∫0∞F(r)g[x3(t-r)]dr,dx2(t)dt=-a2x2(t)+b3f[x3(t-τ3)]+c1∫0∞F(r)g[x1(t-r)]dr,dx3(t)dt=-a3x3(t)+b1f[x1(t-τ1)]+c2∫0∞F(r)g[x2(t-r)]dr,
where F(r)=μ2re-μr  (μ>0) is a strong kernel function, the passive decay rates a1, a2, and a3 are positive constants, bi, ci, and li(i=1,2,3) are constants, and delays τ1, τ2, and τ3 are nonnegative constants. It was emphasized that bifurcating approach is hard to deal with model (8), because it is very difficult to find the bifurcating parameter when τ1, τ2, and τ3 are different nonnegative constants. In order to discuss the existence of oscillating solutions for system (8) we adopt Chafee’s criterion [15]. A time delay system will generate a limit cycle if the system has a unique unstable equilibrium point and bounded solutions. In other words, there exists an oscillatory solution of the model. System (8) can accord with the demands of Chafee’s criterion; we refer the reader to [16, appendix].


## 2. Preliminaries

First we assume that the activation functionsf and g both are monotone continuous bounded functions. f, g∈C4(R), f(0)=g(0)=0, uf(u)>0, and ug(u)>0 for u≠0. For example, f(u)=tanh(u), and g(u)=arctan⁡(u) satisfy those conditions. For the kernel function F(r)=μ2re-μr  (μ>0) we have
(9)∫0∞F(r)g[xi(t-r)]dr=∫0∞μ2re-μrg[xi(t-r)]dr=∫-∞tμ2(t-s)e-μ(t-s)g[xi(s)]ds=μ2e-μt(t∫-∞teμsg[xi(s)]dsffffffffffff-∫-∞tseμsg[xi(s)]ds)(i=1,2,3).
From (9) we get
(10)ddt{∫0∞F(r)g[xi(t-r)]dr}=-μ∫0∞F(r)g[xi(t-r)]dr+μ2e-μt∫-∞teμsg[xi(s)]ds(i=1,2,3).Taking the derivative on both sides of system (8) we obtain
(11)dx12(t)dt2=-a1dx1(t)dt+b2f′[x2(t-τ2)]dx2(t-τ2)dt-μc3∫0∞F(r)g[x3(t-r)]dr+c3μ2e-μt∫-∞teμsg[x3(s)]ds,dx22(t)dt2=-a2dx2(t)dt+b3f′[x3(t-τ3)]dx3(t-τ3)dt-μc1∫0∞F(r)g[x1(t-r)]dr+c1μ2e-μt∫-∞teμsg[x1(s)]ds,dx32(t)dt2=-a3dx3(t)dt+b1f′[x1(t-τ1)]dx1(t-τ1)dt-μc2∫0∞F(r)g[x2(t-r)]dr+c2μ2e-μt∫-∞teμsg[x2(s)]ds.
Set
(12)-μc3∫0∞F(r)g[x3(t-r)]dr=-2μc3∫0∞F(r)g[x3(t-r)]dr+μc3∫0∞F(r)g[x3(t-r)]dr,
so (11) can be rewritten as follows:
(13)dx12(t)dt2=-a1dx1(t)dt+b2f′[x2(t-τ2)]dx2(t-τ2)dt-2μ[dx1(t)dt+a1x1(t)-b2f[x2(t-τ2)]]+μc3∫0∞F(r)g[x3(t-r)]dr+c3μ2e-μt∫-∞teμsg[x3(s)]ds,dx22(t)dt2=-a2dx2(t)dt+b3f′[x3(t-τ3)]dx3(t-τ3)dt-2μ[dx2(t)dt+a2x2(t)-b3f[x3(t-τ3)]]+μc1∫0∞F(r)[gx1(t-r)]dr+c1μ2e-μt∫-∞teμsg[x1(s)]ds,dx32(t)dt2=-a3dx3(t)dt+b1f′[x1(t-τ1)]dx1(t-τ1)dt-2μ[dx3(t)dt+a3x3(t)-b1f[x1(t-τ1)]]+μc2∫0∞F(r)g[x2(t-r)]dr+c2μ2e-μt∫-∞teμsg[x2(s)]ds.Taking the derivative again on both sides of (13) and using (10) give
(14)dx13(t)dt3=-a1dx12(t)dt2+b2f′′[x2(t-τ2)](dx2(t-τ2)dt)2+b2f′[x2(t-τ2)]dx22(t-τ2)dt2-2μdx12(t)dt2-2μa1dx1(t)dt-2μb2f′[x2(t-τ2)]dx2(t-τ2)dt-μ2(dx1(t)dt+a1x1(t)-b2f[x2(t-τ2)])+c3μ2g[x3(t)],dx23(t)dt3=-a2dx22(t)dt2+b3f′′[x3(t-τ3)](dx3(t-τ3)dt)2+b3f′[x3(t-τ3)]dx32(t-τ3)dt2-2μdx22(t)dt2-2μa2dx2(t)dt-2μb3f′[x3(t-τ3)]dx3(t-τ3)dt-μ2(dx2(t)dt+a2x2(t)-b3f[x3(t-τ3)])+c1μ2g[x1(t)],dx33(t)dt3=-a3dx32(t)dt2+b1f′′[x1(t-τ1)](dx1(t-τ1)dt)2+b1f′[x1(t-τ1)]dx12(t-τ1)dt2-2μdx32(t)dt2-2μa3dx3(t)dt-2μb1f′[x1(t-τ1)]dx1(t-τ1)dt-μ2(dx3(t)dt+a3x3(t)-b1f[x1(t-τ1)])+cμ2g[x2(t)].Now by settingx4(t)=dx1(t)/dt, x5(t)=dx2(t)/dt, x6(t)=dx3(t)/dt, x7(t)=dx12(t)/dt2, x8(t)=dx22(t)/dt2, and x9(t)=dx32(t)/dt2, we get the following time delay equivalent system of (8):
(15)dx1(t)dt=x4(t),dx2(t)dt=x5(t),dx3(t)dt=x6(t),dx4(t)dt=x7(t),dx5(t)dt=x8(t),dx6(t)dt=x9(t),dx7(t)dt=-μ2a1x1(t)-(2μa1+μ2)x4(t)-(a1+2μ)x7(t)+b2f′′[x2(t-τ2)]×[x5(t-τ2)]2+b2f′[x2(t-τ2)]x8(t-τ2)-2μb2f′[x2(t-τ2)]x5(t-τ2)+μ2b2f[x2(t-τ2)]+c3μ2g[x3(t)],dx8(t)dt=-μ2a2x2(t)-(2μa2+μ2)x5(t)-(a2+2μ)x8(t)+b3f′′[x3(t-τ3)]×[x6(t-τ3)]2+b3f′[x3(t-τ3)]x9(t-τ3)-2μb3f′[x3(t-τ3)]x6(t-τ3)+μ2b3f[x3(t-τ3)]+c1μ2g[x1(t)],dx9(t)dt=-μ2a3x3(t)-(2μa3+μ2)x6(t)-(a3+2μ)x9(t)+b1f′′[x1(t-τ1)]×[x4(t-τ1)]2+b1f′[x1(t-τ1)]x7(t-τ1)-2μb1f′[x1(t-τ1)]x4(t-τ1)+μ2b1f[x1(t-τ1)]+c2μ2g[x2(t)].
The linearization of system (15) around the zero point is the following:
(16)dx1(t)dt=x4(t),dx2(t)dt=x5(t),dx3(t)dt=x6(t),dx4(t)dt=x7(t),dx5(t)dt=x8(t),dx6(t)dt=x9(t),dx7(t)dt=-μ2a1x1(t)-(2μa1+μ2)x4(t)-(a1+2μ)x7(t)+b2f′(0)x8(t-τ2)-2μb2f′(0)x5(t-τ2)+μ2b2f′(0)x2(t-τ2)+c3μ2g′(0)x3(t),dx8(t)dt=-μ2a2x2(t)-(2μa2+μ2)x5(t)-(a2+2μ)x8(t)+b3f′(0)x9(t-τ3)-2μb3f′(0)x6(t-τ3)+μ2b3f′(0)x3(t-τ3)+c1μ2g′(0)x1(t),dx9(t)dt=-μ2a3x3(t)-(2μa3+μ2)x6(t)-(a3+2μ)x9(t)+b1f′(0)x7(t-τ1)-2μb1f′(0)x4(t-τ1)+μ2b1f′(0)x1(t-τ1)+c2μ2g′(0)x2(t).
System (16) can be written as a matrix form:
(17)dX(t)dt=PX(t)+QX(t-τ~),
where X(t)=(x1(t),x2(t),…,x9(t))T, X(t-τ~) = (x1(t-τ1), x2(t-τ2), x3(t-τ3), x4(t-τ1), x5(t-τ2), x6(t-τ3), x7(t-τ1), x8(t-τ2), and x9(t-τ3))T. So,
(18)P=(000000000100010001000000000000000000000000000100010001p710p73p81p8200p92p93p74000p85000p96p77000p88000p99),Q=(0000000000000000000000000000000000000000000000000000000q72000q83q91000q75000q86q94000q78000q89q9700),
where p71=-μ2a1, p73=c3μ2g′(0), p74=-(2μa1+μ2), p77=-(a1+2μ), p81=c1μ2g′(0), p82=-μ2a2, p85=-(2μa2+μ2), p88=-(a2+2μ), p92=c2μ2g′(0), p93=-μ2a3, p96=-(2μa3+μ2), p99=-(a3+2μ); q72=μ2b2f′(0), q75=-2μb2f′(0), q78=b2f′(0), q83=μ2b3f′(0), q86=-2μb3f′(0), q89=b3f′(0), q91=μ2b1f′(0), q94=-2μb1f′(0), and q97=b1f′(0).Lemma 1.
All solutions of system (8) are bounded.Proof.
Since we assume thatf and g both are monotone continuous bounded functions. So we have |f(u)|≤M and |g(u)|≤N(M and N are positive constants). From (8) we obtain
(19)d|x1(t)|dt≤-a1|x1(t)|+A1,d|x2(t)|dt≤-a2|x2(t)|+A2,d|x3(t)|dt≤-a3|x3(t)|+A3,
where A1=|b2|M+|c3|N, A2=|b3|M+|c1|N, and A3=|b1|M+|c2|N. Thus, |xi(t)|≤(Ai/ai)+|xi(0)|  (i=1,2,3). This means that the solutions of system (8) are uniformly bounded.Lemma 2.
Assume that the matrix(20)B=(-a1b2f′(0)c3g′(0)c1g′(0)-a2b3f′(0)b1f′(0)c2g′(0)-a3)
is a nonsingular matrix. Then system (8) has a unique equilibrium point.Proof.
Noting thatf and g both are monotone continuous bounded functions satisfying f(0)=g(0)=0. Then f(x)=f′(0)x+α, g(x)=g′(0)x+β, where α and β both are higher order infinitesimals when x→0. From (8), an equilibrium point x*=(x1*,x2*,x3*)T is a solution of the following algebraic equation:
(21)-a1x1*+b2f(x2*)+c3∫0∞F(r)g(x3*)dr=0,-a2x2*+b3f(x3*)+c1∫0∞F(r)g(x1*)dr=0,-a3x3*+b1f(x1*)+c2∫0∞F(r)g(x2*)dr=0.
Noting that∫0∞F(r)g(xi*)dr=g(xi*)∫0∞F(r)dr=g(xi*), then we have
(22)-a1x1*+b2f′(0)x2*+b2α+c3g′(0)x3*+c3β=0,-a2x2*+b3f′(0)x3*+b3α+c1g′(0)x1*+c1β=0,-a3x3*+b1f′(0)x1*+b1α+c2g′(0)x2*+c2β=0.
Ify*=(y1*,y2*,y3*)T is another equilibrium point of system (8), neglecting of the higher order infinitesimal, we obtain
(23)-a1(x1*-y1*)+b2f′(0)(x2*-y2*)+c3g′(0)(x3*-y3*)=0,-a2(x2*-y2*)+b3f′(0)(x3*-y3*)+c1g′(0)(x1*-y1*)=0,-a3(x3*-y3*)+b1f′(0)(x1*-y1*)+c2g′(0)(x2*-y2*)0.
The matrix form of (23) is follows:
(24)B(x*-y*)=0.
Since B is a nonsingular matrix based on the algebraic knowledge one can have x*-y*=0 or x*=y*. This means that system (8) has a unique equilibrium point. Obviously, the unique equilibrium point exactly is zero point since f(0)=g(0)=0.


## 3. Oscillating Solutions Analysis

We adopt the following norms of vectors and matrices:∥x(t)∥=∑i=1n|x(t)|,∥P∥=max⁡j∑i=19|pij|,  ∥Q∥=max⁡j∑i=1n|qij|. The measure σ(P) of the matrix P is defined by σ(P)=lim⁡θ→0+⁡((∥I+θP∥-1)/θ), which for the chosen norms reduces to σ(P)=max⁡1≤j≤9(pjj+∑i=1i≠j,9|pij|).In order to discuss the instability of equilibrium point for system (8), we consider the equivalent system (15) of (8). Note that the linearized system of (15) is (16). Obviously, if the trivial solution of (16) is unstable, it implies that the trivial solution of system (15) is unstable and thus the instability of the trivial solution of system (8). Therefore, we first haveTheorem 3.
Assume that system (8) has a unique equilibrium point and the determinant of matrix P is not equal to zero. Let ρ1,ρ2,…,ρ9,ϱ1,ϱ2,…,ϱ9 be the eigenvalues of the matrixes P and Q, respectively. If there is at least one Reρi> 0 (Im⁡ρi may be equal to zero) or there exists one positive real eigenvalue ϱj>|ρj|, for some j∈{1,2,…,9}, then the unique equilibrium point, namely, the trivial solution of system (16) is unstable, implying that the equivalent system (15) or (8) generates oscillating solutions.Proof.
SinceQ is a singular matrix, set ϱ1=ϱ2=⋯=ϱ6=0. For given value of μ, let ρ1,ρ2,…,ρ9 be the eigenvalues of the matrix P. Thus, the characteristic equation of system (16) is as follows:
(25)det⁡(λIij-pij-qije-λτij)=0,
where
(26)Iij={1ifi=j,0ifi≠j,τij={τiifi=j,0ifi≠j.
or
(27)∏i=19[λ-ρi-ϱie-λτi]=0.
Since there is at least oneReρi>0, without loss of generality, set Reρ1>0. From ϱ1=0 we have the equation λ-ρ1=0. This means that λ=ρ1, and there is a Re λ>0. So, the unique equilibrium point of system (16) is unstable based on the theory of differential equation. If by corresponding some ρj we have ϱj>|ρj|  j∈{1,2,…,9}, we pointed out that there exists positive real root for transcendental equation λ-ρj-ϱje-λτj=0. Let f(λ)=λ-ρj-ϱje-λτj. Then f(λ) is a continuous function of λ. Since f(0)=-ρj-ϱj, from ϱj>|ρj|, we know that f(0)<0. Obviously, there is a suitable large λ*>0 such that f(λ*)=λ*-ρj-ϱje-λ*τj>0 since e-λ*τj can be suitably small. Thus, by the continuity of f(λ), there is a point, say λ-∈(0,λ*) such that f(λ-)=0. In other words, there is a positive real eigenvalue. In this case the unique equilibrium point of system (16) is also unstable. Since all solutions of the system are bounded, on the basis of Chafee’s criterion [15], system (15) generates a limit cycle, implying that there exists an oscillating solution of system (15) and therefore system (8).Theorem 4.
Assume that system (8) has a unique equilibrium point and the following inequalities hold:
(28)(∥Q∥τe)exp⁡⁡(-|σ(P)|τ)>1,(∥Q∥τ*e)exp⁡⁡(-τ*|σ(P)|τ*)>1,
where τ=min⁡⁡{τ1,τ2,τ3} and τ*=max⁡{τ1,τ2,τ3}. Then the unique equilibrium point, namely, the trivial solution of system (16) is unstable, implying that the equivalent system (15) or (8) generates oscillating solutions.Proof.
First we consider system (16) in the case τ1=τ2=τ3=τ and we easily get:
(29)d|x1(t)|dt=|x4(t)|,d|x2(t)|dt=|x5(t)|,d|x3(t)|dt=|x6(t)|,d|x4(t)|dt=|x7(t)|,d|x5(t)|dt=|x8(t)|,d|x6(t)|dt=|x9(t)|,d|x7(t)|dt≤-μ2a1|x1(t)|-(2μa1+μ2)|x4(t)|-(a1+2μ)|x7(t)|+|b2f′(0)||x8(t-τ)|+|2μb2f′(0)||x5(t-τ2)|+|μ2b2f′(0)||x2(t-τ2)|+|c3μ2g′(0)||x3(t)|,d|x8(t)|dt≤-μ2a2|x2(t)|-(2μa2+μ2)|x5(t)|-(a2+2μ)|x8(t)|+|b3f′(0)||x9(t-τ3)|+|2μb3f′(0)||x6(t-τ3)|+|μ2b3f′(0)||x3(t-τ3)|+|c1μ2g′(0)||x1(t)|,d|x9(t)|dt≤-μ2a3|x3(t)|-(2μa3+μ2)|x6(t)|-(a3+2μ)|x9(t)|+|b1f′(0)||x7(t-τ)|+|2μb1f′(0)||x4(t-τ1)|+|μ2b1f′(0)||x1(t-τ1)|+|c2μ2g′(0)||x2(t)|.
Let(t)=∑i=19|xi(t)|; then for some t*,y(t)>0  (t≥t*) and we have
(30)dy(t)dt≤σ(P)y(t)+∥Q∥y(t-τ),t≥t*+τ.
Consider the scalar equation(31)dz(t)dt=σ(P)z(t)+∥Q∥z(t-τ),t≥t*+τ
with y(s)=z(s) and s∈[t*,t*+τ]. According to the comparison theorem of differential equation, one can obtain
(32)y(t)≤z(t),t≥t*+τ.
We claim that the trivial solution of (31) is unstable. Suppose that this is not true; then the characteristic equation associated with (31) given by
(33)λ=σ(P)+∥Q∥e-λτ
will have a real negative root say λ* such that λ*=σ(P)+∥Q∥e-λ*τ, where e-λ*τ=e|λ*τ|.
So we get(34)|λ*|≥∥Q∥e|λ*τ|-|σ(P)|.
Thus
(35)1≥∥Q∥e|λ*τ||λ*|+|σ(P)|=∥Q∥τexp⁡⁡(-|σ(P)|τ)exp⁡⁡(|λ*|τ+|σ(P)τ|)|λ*|τ+|σ(P)τ|.
Based on the formulaex≥ex(x>0), from (35), we have
(36)1≥(∥Q∥τe)exp⁡⁡(-|σ(P)|τ).Inequality (36) contradicts the first inequality of (28). Therefore, our claim regarding the instability of the trivial solution is valid. Similarly, consider system (16) in the case τ1=τ2=τ3=τ*; we know that the trivial solution is unstable if the second inequality of (28) holds. Note that τ≤τi≤τ*(i=1,2,3). So, the trivial solution of (16) is unstable when condition (28) holds, implying that the trivial solution of (15) is unstable. According to Chafee’s criterion, system (15) generates a limit cycle, suggesting that there is an oscillating solution of system (15) and therefore system (8).


## 4. Computer Simulations

We use the equivalent system (15) of (8) for computer simulation. In Figure 1 both activation functions f(u) and g(u) are taken tanh (u). Thus, f′(u)=g′(u)=4/(eu+e-u)2=1-(tanh⁡(u))2, f′′(u)=g′′(u)=2(tanh⁡(u))3-2tanh⁡(u), and f′(0)=g′(0)=1. We select μ=5.5, a1=0.48, a2=0.65, a3=0.78, b1=0.6, b2=0.8, b3=0.9, c1=-0.8, and c2=-1.8, c3=0.4. Then the eigenvalues of matrix P are -5.8250±1.8620i, -1.7568±2.5151i, −0.0000, -0.4840,-5.3568,-5.6392, and −8.2087 and the eigenvalues of matrix Q are -0.3780±0.6547i, 0.7560, and 0,0,0,0,0,0. Note that the eigenvalue 0.7560 of matrix Q is larger than |-0.4840|=0.4840. Based on Theorem 3, the trivial solution of system (15) is unstable. The system generates an oscillating solution. In order to compare the effect of the time delays, we change the delays from (0.2, 0.3, and 0.4) to (1, 1.5, and 2), while keeping the other parameters as shown in Figure 1. We see that the oscillatory amplitude and frequency both are changed (see Figure 2). In Figure 3, we keep the parameter values the same as Figure 2, the activation function g(u)=tanh⁡(u), and change the activation function f(u)=arctan⁡(u). Then f′(u)=1/(1+u2) and f′′(u)=-2u/(1+u2)2. We see that the graph is changed slightly in comparison with Figure 2. In Figure 4, we set the activation function f(u)=1/(1+e-2u) and g(u)=(e-2u-1)/(1+e-2u). Then f′(u)=2e-2u/(1+e-2u)2, g′(u)=-4e-2u/(1+e-2u)2, f′′(u)=4e-2u(2e-2u-1)/(1+e-2u)3, and g′′(u)=8e-2u(1-e-2u)/(1+e-2u)3. Thus f′(0)=1/2 and g′(0)=-1. Note that f(0)=1/2≠0 and g(0)=-1/2≠0. When we select the parameters as follows: μ=3.5, a1=0.18, a2=0.15, a3=0.19, b1=0.05, b2=0.08, b3=0.12, c1=-0.6, and c2=0.8, c3=0.5, time delays are (0.5, 0.6, and 0.7). The oscillatory solutions also appeared. This means that the restrictive condition f(0)=g(0)=0, uf(u)>0, and ug(u)>0 for u≠0 is only for convenience of the proof.Oscillatory behavior of the solutions:mu=3.5, delays: (0.5, 0.6, and 0.7), and activation functions: f(u)=g(u)=tanh(u).
(a)
solid line:x1(t), dashed line: x2(t), and dotted line: x3(t)
(b)
solid line:x4(t), dashed line: x5(t), and dotted line: x6(t)
(c)
solid line:x7(t), dashed line: x8(t), and dotted line: x9(t)Oscillatory behavior of the solutions:mu=5.5, delays: (1, 1.5, and 2), and activation functions: f(u)=g(u)=tanh(u).
(a)
solid line:x1(t), dashed line: x2(t), and dotted line: x3(t)
(b)
solid line:x4(t), dashed line: x5(t), and dotted line: x6(t)
(c)
solid line:x7(t), dashed line: x8(t), and dotted line: x9(t)Oscillatory behavior of the solutions:mu=5.5, delays: (1, 1.5, and 2), and activation functions: f(u)=arctan(u), g(u)=tanh(u).
(a)
solid line:x1(t), dashed line: x2(t), and dotted line: x3(t)
(b)
solid line:x4(t), dashed line: x5(t), and dotted line: x6(t)
(c)
solid line:x7(t), dashed line: x8(t), and dotted line: x9(t)Oscillatory behavior of the solutions:mu=3.5, delays: (0.5, 0.6, and 0.7), and activation functions: f(u)=1/(1+exp⁡(-2u)), g(u)=(exp⁡(-2u)-1)/(1+exp⁡(-2u)).
(a)
solid line:x1(t), dashed line: x2(t), and dotted line: x3(t)
(b)
solid line:x4(t), dashed line: x5(t), and dotted line: x6(t)
(c)
solid line:x7(t), dashed line: x8(t), and dotted line: x9(t)


## 5. Conclusion

This paper discusses the oscillatory behavior of the solutions for a three-node network model with discrete and distributed delays. Two theorems are provided to ensure the existence of oscillating solutions for the model. Computer simulations suggested that our theorems are only sufficient conditions. How to find a necessary condition is still an open problem.



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*Source: 536324-2014-02-24.xml*