Analytical and numerical stability analysis ofnonlinear generalized delay differentialequations with variable delay terms

JIANG Chengxiang

(Tianhua College,Shanghai Normal University,Shanghai 201815,China)

1 Introduction

Delay differential equations provide a powerful model of many phenomena in applied sciences such as physics,engineering,biology, and economics.In this paper,we consider the stability behavior of the analytical and numerical solutions of the system of generalized delay differential equation with variable delay terms:

(1)

wheref:[0,+∞)×Cd×Cd→Cd,φdenote given complex functions, andy(tτ)=(y1(t-τ1(t)),y2(t-τ2(t)),…,yd(t-τd(t)))T,andτi(t) (i=1,2,…,d) are continuous differential functions satisfing the following hypothese:

(H1)τi(t)≥τj(t)≥τ0>0(i>j),for allt≥t0.

The stability properties of the numerical methods for linear delay differential equations have been widely studied by many authors[1-6].In [7-9],Cong etc.investigated the stability properties of numerical methods for linear generalized delay differential equation with a variable lag (GDDEs).Because of the complexity of nonlinearGDDEs,there were no papers dealing with it.

In this paper,a sufficient condition for the asymptotical stability of the theoretical solution of (1) is discussed.Then,we investigate the numerical stability of Runge-Kutta methods for systems ofGDDEs.A numerical test at the end of this paper confirms our theoretical results.

2 Stability of analytical solution

(2)

(3)

wherefis a given mapping which satisfies the following conditions:

∀t≥t0,y1,y2,u∈Cd.

(4)

(5)

whereα(t),β(t) are continuous bounded functions.

(6)

Lemma2.1[11]Ifv(t)>0,t∈(-∞,+∞),and

(7)

whereψ(t) is continuous and bounded fort≤t0,A(t),B(t)≥0 fort∈ [t0,+∞)],τ(t)≥0 andt-τ(t)→+∞,ast→+∞,and if there exits aσ>0 such that

-A(t)+B(t)≤-σ<0, fort≥t0,

thenv(t)→0,ast→∞.

Theorem2.1If the systems ofGDDEs(2) and (3) satisfy (4),(5) andα(t)<0,∀t≥t0,and

α(t)+β(t)≤-σ<0,

(8)

then the system is asymptotically stable.

Thus we have completed the proof.

3 Numerical stability analysis

We now investigate the stability analysis of the (k,l)-algebraically stable Runge-Kutta methods for nonlinearGDDEs.

Now we consider the adaptation of thes-stage Runge-Kutta methods to (2).

(9)

Similarly,the adaptation of the Runge-Kutta Methods with the same interpolation procedure for the problem (3) leads to the following process:

(10)

Let

(11)

It follows from (9) and (10),that

(12)

Definition3.1Letlbe a real constant.A Runge-Kutta method with aninterpolation procedure is said to beGAR(l)-stable if

(13)

with stepsizehsatisfying (α+β)≤l.

Definition3.2[10]Letk,lbe real constants.An RK method is said to be (k,l)-algebraically stable if there exists a diagonal nonnegative matrixGandD=diag(d1,…,ds) such thatM=(mij) is nonnegative,where

In this paper,we use the linear interpolation procedure.Letτi(tn+cjh)=(imj(n)-iδj(n))hwith integerimj(n)≥1 andδj(n)∈[0,1).

Let

(14)

Theorem3.1Assume that a RK method is (k,l)-algebraically stable,then

(15)

ProofIt is well known[10]that

(16)

Because of the (k,l)-algebraically stability of the method,we have:

(17)

It follows from (4),(5) and (16) that

Theorem3.2Assume that a Runge-Kutta method is (k,l)-algebraically stable andk<1.Then the method with linear interpolation procedure isGAR(l)-stable.

ProofLet

μ=(2α+β)h-2l,

and

The application of Theorem 3.1 yields

By induction,we have

On the other hand,

which shows that the method isGAR(l)-stable.

4 Numerical experiment

We use the classical Runge-Kutta method of order 4 to solveGDDEsfor confirming the theoretical results.

Consider the following generalized delay differential equation:

(18)

and its perturbed problem

(19)

Table 1 Error compared to the computing time t of the RK method for the above equations

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