Computing Stability of Differential Equations with Bounded Distributed Delays

Numerical Algorithms 34: 41–66, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands.

Computing stability of differential equations with boundeddistributed delays

T. Luzyanina ∗, K. Engelborghs and D. RooseDepartment of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200 A,

B-3001 Heverlee-Leuven, BelgiumE-mail: [email protected]

Received 10 June 2002; accepted 21 May 2003Communicated by K. Burrage

This paper deals with the stability analysis of scalar delay integro-differential equations(DIDEs). We propose a numerical scheme for computing the stability determining character-istic roots of DIDEs which involves a linear multistep method as time integration scheme anda quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule.We investigate to which extent the proposed scheme preserves the stability properties of theoriginal equation. We derive and prove a sufficient condition for (asymptotic) stability of aDIDE (with a constant kernel) which we call RHP-stability. Conditions are obtained underwhich the proposed scheme preserves RHP-stability. We compare the obtained results withcorresponding ones using Newton–Cotes formulas. Results of numerical experiments on com-puting the stability of DIDEs with constant and nonconstant kernel functions are presented.

Keywords: delay integro-differential equations, quadrature rules, numerical stability analysis

AMS subject classification: 65P30, 65J15

1. Introduction

This paper aims at the numerical stability analysis of the zero solution of the fol-lowing differential equation with bounded and continuously distributed delay

d

dty(t) = A0y(t) + A1

∫ t−τ1

t−τ2

K(t − ξ)y(ξ) dξ, τ2 > τ1 � 0, (1.1)

which is usually called a delay integro-differential equation (DIDE). Equation (1.1) canbe considered as the linearization of a nonlinear DIDE around its steady state solution(equilibrium). Nonlinear DIDEs appear, e.g., in modelling population dynamics, thespread of infectious diseases and control theory (e.g., [1,12,13,18,20,21]). The local sta-bility and bifurcation analysis of a steady state solution of these models is one of the

∗ On leave from the Institute of Mathematical Problems in Biology, Pushchino, Moscow region, 142290,Russia.

42 T. Luzyanina et al. / Stability of differential equations with distributed delays

primary objectives in the models’ analysis and it is achieved through the linearization ofa nonlinear DIDE around this solution, i.e. through equation (1.1).

Our goal is a numerical method which allows to approximate the rightmost (sta-bility determining) roots of the characteristic equation for (1.1). We extend the methodsdescribed in [9,10] for equations with discrete delays to DIDEs. The characteristic rootsdetermine the (local) stability of the steady state solution and are of interest to locate andidentify bifurcations. The method proposed here consists of a time integration schemefor (1.1) which is used to construct a matrix-approximation of the time integration op-erator of (1.1) over a single time step. The eigenvalues of this matrix determine the(asymptotic) stability of the zero solution of the discretized equation and approximate(a finite number of) the eigenvalues of the solution operator. Hence, to ensure correct re-sults, we are interested in schemes preserving (under certain conditions) the (asymptotic)stability of the original equation. Note that, since we do not perform time integration,it is not our goal to find a numerical scheme for efficient time integration of DIDEs.The fact that we need a mapping between the meshes before and after a single time stepalready leads to the restriction of a fixed steplength (see [10] for more details). Never-theless stability results obtained in the context of time integration have a direct impacton the issues we consider and so we briefly review existing results in this direction.

The numerical treatment of equations with distributed delays is more complicatedcompared to the case of discrete delays due to the integral terms. In particular, it is knownthat the kernel function not only affects the stability of solutions but also the stabilityproperties of the numerical scheme used. A few results in this direction have appeared.In [3], the stability properties of a numerical scheme based on a linear multistep (LMS)method and a Newton–Cotes formula approximating a scalar DIDE with τ1 = 0 andK(ξ) ≡ 1, ξ ∈ [0, τ2], are discussed. The authors compare the analytically obtainedstability region in the (A0, A1)-plane with the one obtained numerically by employingthe method of D-partitions and the boundary locus technique (see, e.g., [2]). In [19], thestability of Runge–Kutta methods is studied on the basis of a system of equations similarto (1.1) (τ1 = 0, K(ξ) ≡ 1, ξ ∈ [0, τ2], with an additional delay term, x(t − τ2)).It is shown that every A-stable Runge–Kutta method preserves the delay-independentstability of the original equation whenever a step size h = τ2/m is used, m ∈ N.

In this paper we restrict ourselves to scalar DIDEs (A0, A1 ∈ R0) with commen-surate (rationally dependent) delays τ1 and τ2. Note that the latter is trivially true whenτ1 = 0. The considered delays constitute a more general class of delays than the caseτ1 = 0 (which has been analyzed in the literature so far) and provide the necessary in-sight for a future generalization to the case of arbitrary delays. In our analysis, we alsoassume that the kernel function K, defined on the interval [τ1, τ2], is bounded and hassuitable continuity properties.

The numerical scheme proposed in this paper to approximate (1.1) involves anLMS method as time integration scheme and a quadrature method based on Lagrangeinterpolation and a Gauss–Legendre quadrature rule. The choice of these methods ismotivated as follows. The use of LMS methods for stability analysis of systems of dif-ferential equations with multiple discrete delays (DDEs) was investigated in [10] and

T. Luzyanina et al. / Stability of differential equations with distributed delays 43

effectively used in [7,8]. The use of Runge–Kutta methods for time integration of DDEsis often preferred to LMS methods because they extend better to an adaptive steplengthprocedure, especially when taking into account the derivative discontinuities present inthe transient phase of DDE trajectories. However, these arguments do not hold for thesituation under consideration since we are necessarily restricted to fixed steplengths andsince our analysis concerns the asymptotic rather than the transient behavior. Moreover,for stability reasons, the use of Runge–Kutta methods for DDEs is combined with eq-uistage interpolation for evaluation of the delayed terms [14,15] which implies a largerinterpolation interval when compared to LMS methods. The latter can lead to muchlarger error constants in the fixed steplength case as has been shown in [6]. The useof Lagrange interpolation to approximate the delayed terms within an LMS scheme hasshown to be efficient with respect to convergence and stability properties of the resultingnumerical scheme (and is for this reasons preferred to the natural dense output interpo-lation of LMS methods, see [24,25]). Gaussian quadrature rules (“the only all-purposeformulas” [23]) are widely used in applications due to their convergence properties andapplicability to integrands having only low-order derivatives.

We derive and prove a sufficient condition for the asymptotic stability of a DIDEwith a constant kernel function, which we call RHP-stability. We investigate stabilityproperties of the proposed numerical scheme with respect to preserving RHP-stabilityand compare with corresponding results for Newton–Cotes formulas. We show thatRHP-stability is preserved, under certain conditions, by the proposed scheme, whileit is not preserved when using Newton–Cotes formulas. Even when these conditions arenot fulfilled, the proposed scheme remains preferable to Newton–Cotes formulas withrespect to preserving stability, as we show numerically. We present results of numericalexperiments on computing stability of (1.1) for both constant and nonconstant kernelfunctions.

The paper is structured as follows. In section 2 we obtain discretized equationsapproximating (1.1). In section 3 we introduce RHP-stability and formulate conditionsunder which the discretized equation retains the RHP-stability of the original equation.The conditions related to quadrature are further investigated for the proposed quadratureand for Newton–Cotes formulas in section 4. Results of numerical experiments on com-puting stability and characteristic roots of (1.1) are given in section 5. We conclude insection 6. Appendix A contains the proofs of our theorems.

2. Numerical approximation of equation (1.1)

2.1. Quadrature rules

To approximate the integral in (1.1), we propose a quadrature method based on La-grange interpolation and Gauss–Legendre quadrature rules. We also formulate Newton–Cotes formulas which we use for comparison.


2.1.1. Proposed quadrature methodAssume that τ1 and τ2 are commensurate. Then, let h > 0 be a (constant) step size

in the discretization of (1.1) such that τ1 = n1h and τ2 = n2h, n1 ∈ N, n2 ∈ N0. Setn := n2 − n1. Let yj be the numerical approximation of y(t) at the mesh point tj = jh

(assuming t0 = 0).The integral in (1.1) can be written in the form∫ tj−τ1

tj−τ2

K(tj − ξ)y(ξ) dξ =n∑

r=1

∫ h

0K(q1,rh − ξ)y(tj−q1,r + ξ) dξ, (2.1)

where q1,r = n2 − r + 1. Using ξ = εh, ε ∈ [0, 1], we approximate the solutiony(tj−q1,r + ξ) by the Lagrange interpolation polynomial of degree s− + s+,

y(tj−q1,r + εh) s+∑

l=−s−

Pl(ε) yj−q1,r+l , Pl(ε) :=s+∏

ν=−s−,ν �=l

ε − ν

l − ν, (2.2)

through the points at j −q1,r −s−, . . . , j −q1,r +s+, where s−, respectively s+ representthe number of interpolation points taken to the left, respectively to the right of tj−q1,r .Then, we substitute (2.2) in (2.1) and apply the m-point Gauss–Legendre quadratureformula (see, e.g., [23]). The result reads∫ tj−τ1

tj−τ2

K(tj − ξ)y(ξ) dξ n∑

r=1

m∑k=1

BkK(q1,rh − εkh)

s+∑l=−s−

Pl(εk)yj−q1,r+l , (2.3)

where εkh and Bk are respectively the zeros of the mth-degree Legendre polynomial(Gaussian points) and the weights of the Gauss–Legendre quadrature rule on [0, h]. ForK(ξ) ≡ 1, ξ ∈ [τ1, τ2], we will use a compact form of (2.3),∫ tj−τ1

tj−τ2

y(ξ) dξ n∑

r=1

s+∑l=−s−

ωlyj−q1,r+l , ωl :=m∑

k=1

BkPl(εk). (2.4)

Note that the Gauss–Legendre quadrature rule is exact for the polynomials used when-ever s− + s+ � 2m− 1. Below we always assume that the latter holds. We avoid the useof future mesh points in (2.3) and in (2.4) if s+ − q1,n � 0, i.e. τ1 � (s+ − 1)h. Notethat the latter implies s+ = 1 if τ1 = 0.

If τ1 < (s+ − 1)h (hence s+ > 1 is permitted in case τ1 = 0), we split the inte-gral (2.1) as∫ tj−τ1

tj−τ2

K(tj −ξ)y(ξ) dξ =∫ tj−(s+−1)h

tj−τ2

K(tj −ξ)y(ξ) dξ+∫ tj−τ1

tj−(s+−1)hK(tj −ξ)y(ξ) dξ.

(2.5)Consider the second integral in the right-hand side of (2.5),∫ tj−τ1

tj−(s+−1)hK(tj − ξ)y(ξ) dξ =

s+−1−n1∑r=1

∫ h

0K(q2,rh − ξ)y(tj−q2,r + ξ) dξ, (2.6)


where q2,r = s+ − r. Since q2,r < s+, we approximate the solution y(tj−q2,r + ξ),using ξ = εh, ε ∈ [0, 1], by the following Lagrange interpolation polynomial of degrees− + s+,

y(tj−q2,r + εh) s(r)+∑

l=−s(r)−

Pl,r (ε) yj−q2,r+l , Pl,r (ε) =s(r)+∏

ν=−s(r)−,ν �=l

ε − ν

l − ν,

s(r)− := s− + r, s(r)+ := s+ − r,

(2.7)

through the points at j − q2,r − s(r)−, . . . , j − q2,r + s(r)+. Note the difference ininterpolation formulas (2.2) and (2.7): in the latter case the interpolation has a rather“one side” nature. For example, when τ1 = 0, the solution y(tj−1 + ξ) is approximatedusing the solution points at j − s− − s+, . . . , j .

After substitution of (2.7) in (2.6) and applying the m-point Gauss–Legendrequadrature rule (also assuming 2m − 1 � s− + s+), the result reads∫ tj−τ1

tj−(s+−1)hK(tj − ξ)y(ξ) dξ

s+−1−n1∑r=1

m∑k=1

BkK(q2,rh − εkh)

s(r)+∑l=−s(r)−

Pl,r (εk)yj−q2,r+l ,

(2.8)and, for K(ξ) ≡ 1, ξ ∈ [τ1, τ2],∫ tj−τ1

tj−(s+−1)hy(ξ) dξ

s+−1−n1∑r=1

s(r)+∑l=−s(r)−

ωl,ryj−q2,r+l , ωl,r :=m∑

k=1

BkPl,r(εk). (2.9)

2.1.2. Newton–Cotes formulasLet m (m � 2) and ωl (l = 1, . . . , m) be the number of points, respectively weights

used in a Newton–Cotes formula (see, e.g., [16]). For a fixed m, let h, the step size ofdiscretization of (1.1), be such that we could use the m-point repeated Newton–Cotesrule on the interval of integration, i.e. τ2 − τ1 = nm(m − 1)h, τ1 = n1,mh, τ2 = n2,mh,n1,m ∈ N, nm, n2,m ∈ N0. Set hm := (m − 1)h. The m-point Newton–Cotes formulaapplied to the integral in (1.1) then leads to∫ tj−τ1

tj−τ2

K(tj − ξ)y(ξ) dξ =nm∑r=1

∫ hm

0K(qrh − ξ)y(tj − qrh + ξ) dξ

nm∑r=1

m∑l=1

ωlK((qr − l + 1)h)yj−qr+l−1, (2.10)

where qr = n2,m − (r − 1)(m− 1) and the weights ωl, l = 1, . . . , m, are defined on theinterval [0, hm].

Note that for the trapezoidal rule m = 2 and ω1 = ω2 = h/2. For a constant kernelfunction, this rule is equivalent to the 2-point Gauss–Legendre quadrature rule (2.4) withs− = 0, s+ = 1 since P0(ε1) + P0(ε2) = P1(ε1) + P1(ε2) = 1 and B1 = B2 = h/2.


2.2. Quadrature within an LMS scheme

Finally, we discretize (1.1) by applying a linear k-step (consistent and irreducible)formula (see, e.g., [11]) coupled with the proposed quadrature. The result reads

k∑j=0

αjyu+j = h

k∑j=0

βj

(A0yu+j + A1

(nc∑r=1

m∑i=1

BiK(q1,rh − εih)

s+∑l=−s−

Pl(εi)yu+j−q1,r+l

+ns∑r=1

m∑i=1

BiK(q2,rh − εih)

s(r)+∑l=−s(r)−

Pl,r (εi)yu+j−q2,r+l

)), (2.11)

where αj , βj are the coefficients of the LMS method, nc = n, ns = 0 if τ1 � (s+ − 1)hand nc = n2 − s+ + 1, ns = s+ − 1 − n1 otherwise.

Similarly, the use of the m-point Newton–Cotes formula within an LMS schemeleads to

k∑j=0

αjyu+j = h

k∑j=0

βj

(A0yu+j + A1

(nm∑r=1

m∑l=1

ωlK((qr − l + 1)h

)yu+j−qr+l−1

)).

(2.12)All notations in (2.11) and (2.12) are defined as in sections 2.1.1, respectively 2.1.2.

Each of these discretized equations defines an approximation of the time integra-tion operator to (1.1) over the time step h. This approximation (a matrix) is a linear mapbetween [yu−L yu−L+1 · · · yu+k−1] and [yu−L+1 yu−L+2 · · · yu+k] (here L := n2 + s−)which is constructed using the discretized equation for yu+k and a shift for all other vari-ables. The eigenvalues of this matrix determine the stability of the zero solution of thediscretized equation and approximate (a finite number of) eigenvalues of the solutionoperator. Comparing the stability of (1.1) and its discrete variants is the subject of therest of the paper.

3. Asymptotic stability of equation (1.1) and its numerical approximations

In this section we present the characteristic equation for (1.1) and the discretizedequations (2.11) and (2.12). We introduce RHP-stability as a sufficient condition for(asymptotic) stability of (1.1) and obtain conditions under which the discretized equationretains the RHP-stability of (1.1).

First we give formal definitions of asymptotic stability of (1.1) and its numericalapproximation taken from [3]. Let C := C[−τ2, 0] be the Banach space of continuousfunctions mapping the interval [−τ2, 0] into R and BC[0,∞) be the space of functionswhich are continuous and bounded on [0,∞). Also let y(t) = ψ(t), t ∈ [−τ2, 0], andδψ ∈ C be a perturbation of ψ which causes a change y → y + δy in the solution y(t),t > 0.


Definition [3]. Equation (1.1) is (asymptotically) stable if, whenever δψ ∈ C we haveδy ∈ BC[0,∞) and δy(t) → 0 as t → ∞. The discretized equations (2.11), (2.12)are (asymptotically, or strictly) stable if arbitrary bounded changes in the starting values(yν , ν = 1, . . . , k − 1) and arbitrary bounded changes δψ ∈ C yield uniformly boundedδyv and δyv → 0 as v → ∞.

The following results give conditions for stability. Equation (1.1) is asymptoticallystable (equivalently, the zero solution of (1.1) is asymptotically stable) if and only if allthe zeros (characteristic roots) of the characteristic equation

λ = A0 + A1

∫ τ2

τ1

K(ξ)e−λξ dξ (3.1)

have negative real part (see, e.g., [3,12]). Note that there exist an infinite number ofroots λ ∈ C of (3.1) and the number of roots in any right half-plane �(λ) > η, η ∈ R,is finite [12]. Hence, the stability is always defined by a finite number of roots, as in thecase of differential equations with discrete delays.

The discretized equation (2.11) is strictly stable (equivalently, its zero solution isstrictly stable) if and only if the characteristic equation

1

h

∑kj=0 αjµ

j∑kj=0 βjµj

=A0 + A1

(nc∑r=1

m∑i=1

BiK(q1,rh − εih)

s+∑l=−s−

Pl(εi)µ−(q1,r−l)

+ns∑r=1

m∑i=1

BiK(q2,rh − εih)

s(r)+∑l=−s(r)−

Pl,r(εi)µ−(q2,r−l)

)(3.2)

has all its roots µ in the open disk |µ| < 1 (see, e.g., [3]).To compare the roots µ of (3.2) with the solutions λ of the characteristic equa-

tion (3.1), we use the relation µ = exp(λh) and we set

LMS(λh) :=∑k

j=0 αjeλhj∑kj=0 βjeλhj

.

Then (3.2) is equivalent to

1

hLMS(λh)=A0 + A1

(nc∑r=1

m∑i=1

BiK(q1,rh − εih)

s+∑l=−s−

Pl(εi)e−λh(q1,r−l)

+ns∑r=1

m∑i=1

BiK(q2,rh − εih)

s(r)+∑l=−s(r)−

Pl,r (εi)e−λh(q2,r−l)

). (3.3)

By the requirement of irreducibility and the fact that solutions µ = 0 do not influencestability, it follows that the stability of the discretized equation is now determined bythe real parts of the (finite number of) solutions λ of (3.3) just as for the characteristicequation (3.1). In the next section we investigate properties of quadrature rules in case


of a constant kernel function. Therefore, we now assume K(ξ) ≡ 1, ξ ∈ [τ1, τ2]. Thestability of equation (1.1) with nonconstant kernel functions is considered in section 5.

We compare the characteristic equations

λ = A0 + A1

∫ τ2

τ1

e−λξ dξ, (3.4)

and

1

hLMS(λh) = A0 + A1

(nc∑r=1

s+∑l=−s−

ωle−λh(q1,r−l) +

ns∑r=1

s(r)+∑l=−s(r)−

ωl,re−λh(q2,r−l)

). (3.5)

Let C+0 , C

+ denote the open, respectively closed right half-plane,

C+0 = {

λ ∈ C | �(λ) > 0}

and C+ = {

λ ∈ C | �(λ) � 0},

and C−0 , C

− denote the open respectively closed left half-plane (with correspondingdefinitions). We define the set-valued functions *c(·) and *ch(·) as

*c(C)=⋃λ∈C

(A0 + A1

∫ τ2

τ1

e−λξ dξ

), C ⊂ C, (3.6a)

*ch(C)=⋃λ∈C

(A0 + A1

(nc∑r=1

s+∑l=−s−


ns∑r=1

s(r)+∑l=−s(r)−


)).

(3.6b)

In the context of stability analysis, the mapping of C+ under *c and *ch, i.e. the sets

*c(C+) and *ch(C

+), is of interest. Note that these sets are bounded subsets of C

because

λ̃ ∈ *c

(C

+) ⇒ |λ̃ − A0| �∣∣∣∣A1(e−λτ1 − e−λτ2)

λ

∣∣∣∣ � |A1|(τ2 − τ1), λ ∈ C+,

λ̃ ∈ *ch

(C

+) ⇒ |λ̃ − A0| � |A1|(nch + nsh) = |A1|(τ2 − τ1).

(3.7)The latter holds because q1,r − s+ � 0, r = 1, . . . , nc,

∑s+l=−s− ωl = h and q2,r − s(r)+

= 0,∑s(r)+

l=−s(r)− ωl,r = h, r = 1, . . . , ns . It follows from (3.7) that the mapping of C+

under *c and *ch is inside a circle centered at A0 with radius |A1|(τ2 − τ1).The condition

*c

(C

+) ⊂ C−0 (3.8)

is a sufficient condition for (asymptotic) stability of (3.4). Indeed, it implies that theright-hand side of (3.4) maps the right half-plane (RHP) �(λ) � 0 into the left half-plane. Hence (3.4) cannot have solutions with positive real part. We will refer to (3.8)as RHP-stability. Obviously, the condition

A0 + |A1|(τ2 − τ1) < 0 (3.9)

is a sufficient condition for RHP-stability.


We will call the region C\LMS(C+) the stability region of the corresponding LMSmethod [11]. It is the region into which LMS(·) does not map any unstable λ, �(λ) � 0.If

*c

(C

+) ⊂ 1

h

(C \ LMS

(C

+)) (3.10a)

and

*ch

(C

+) ⊆ *c

(C

+), (3.10b)

then RHP-stability is recovered by the numerical scheme and hence the zero solution ofthe discretized equation is asymptotically stable.

If C\LMS(C+) contains a semicircle of radius ρLMS, then one can assure condition(3.10a) by an appropriate choice of the step size h,

h <ρLMS

|A0| + |A1|(τ2 − τ1). (3.11)

Note that such a semicircle exists, for instance, for backward differentiation and Adamsmethods with k < 7.

Condition (3.10b) is equivalent to the condition

*h

(τ1, τ2;C

+) ⊆ *(τ1, τ2;C

+), (3.12)

which we study in the next section. Here the set-valued function *(·, ·; ·) is defined as

*(τ1, τ2;C) =⋃λ∈C

(∫ τ2

τ1

e−λξ dξ

), C ⊂ C,

and *h(·, ·; ·) is its discrete analogue. The equivalence of (3.10b) and (3.12) is becauseA0 and A1 represent a translation respectively scaling of the mapping. To distinguish be-tween quadratures, we will replace the notation *h by *h,GL for the proposed quadratureand by *h,NC for Newton–Cotes formulas. So,

*h,GL(τ1, τ2;C)=⋃λ∈C

(nc∑r=1

s+∑l=−s−


ns∑r=1

s(r)+∑l=−s(r)−


), (3.13)

*h,NC(τ1, τ2;C)=⋃λ∈C

(nm∑r=1

m∑l=1

ωle−λh(qr−l+1)

), C ⊂ C. (3.14)

Note that *h,NC(τ1, τ2;C+) was obtained by using a similar analysis for the dis-

cretized equation (2.12) with K = 1. Clearly, *(τ1, τ2;C+), *h,GL(τ1, τ2;C

+) and*h,NC(τ1, τ2;C

+) are bounded subsets of C.We need the following results related to (3.12).

Theorem 3.1. The region defined by *(τ1, τ2;C+) is located inside the curve *(τ1, τ2;⋃

θ∈Riθ).


Due to this theorem, RHP-stability (3.8) is equivalent to the condition

�(A0 + A1

e−iτ1θ − e−iτ2θ

iθ

)< 0 for all θ ∈ R, (3.15)

which implies that for analyzing RHP-stability of the zero solution of (1.1) it is sufficientto examine the mapping of the imaginary axis under *c. Obviously, if (3.15) holds forcertain values of A0, A1, τ1 and τ2, then it also holds for A0, A1/ρ, τ1ρ, τ2ρ, with ρ > 0.Hence RHP-stability is independent of the factor ρ.

The next two theorems are analogous to theorem 3.1.

Theorem 3.2. The region defined by *h,GL(τ1, τ2;C+) is located inside the curve

*h,GL(τ1, τ2;⋃θ∈Riθ).

Theorem 3.3. The region defined by *h,NC(τ1, τ2;C+) is located inside the curve

*h,NC(τ1, τ2;⋃θ∈Riθ).

In the next section we investigate whether condition (3.12) is fulfilled for the pro-posed quadrature scheme and for Newton–Cotes formulas.

4. Properties of quadrature rules for a constant kernel

4.1. Proposed quadrature method

Our study has shown that Lagrange interpolation which we use when τ1 �(s+ − 1)h and τ1 < (s+ − 1)h, cf. (2.2), respectively (2.7), affects the properties ofthe proposed quadrature scheme. Therefore, we consider these two cases separately.

Case τ1 � (s+ − 1)h. Note that τ1 = 0 implies s+ = 1. It follows from (3.13), usingnc = n and ns = 0 (i.e. τ1 � (s+ − 1)h), that

*h,GL(τ1, τ2;C) =⋃λ∈C

(e−λτ1

n∑r=1

e−λh(r−1)s+∑

l=−s−

ωleλh(l−1)

). (4.1)

Theorem 4.1. If s− � τ1/h and s+ = s− + 1, then *h,GL(τ1, τ2;C+) ⊆ *(τ1, τ2;C

+).

Obviously, when τ1 = 0, the conditions of this theorem are only satisfied ifs− = 0 and s+ = 1. The result of the theorem is illustrated in figure 1. Note that in allfigures we use notations ∂*(·, ·) and ∂*h,GL(·, ·) for the boundaries of *(·, ·;⋃θ∈R

iθ),respectively *h,GL(·, ·;⋃θ∈R

iθ).


(a) (b)

Figure 1. Solid and dashed lines denote the boundary curves ∂*, respectively ∂*h,GL. (a) ∂*(0, h) and∂*h,GL(0, h); (b) ∂*(τ1, τ2) and ∂*h,GL(τ1, τ2). h = 1, τ1 = 2, τ2 = 4, s− = 1, s+ = 2.

Case τ1 < (s+ − 1)h. We will assume that s+ = s− + 1 (hence τ1 < s−h) and thats− > n1 (since s− � n1 ⇒ τ1 � s−h). It follows from (3.13), using nc = n2 − s− andns = s− − n1, that

*h,GL(τ1, τ2;C)=⋃λ∈C

(e−λhs−

n2−s−∑r=1

e−λh(r−1)s+∑

l=−s−

ωleλh(l−1)

+s−−n1∑r=1

e−λh(s+−r)

s(r)+∑l=−s(r)−

ωl,reλhl

). (4.2)

Numerical experiments have shown that whether or not *h,GL(τ1, τ2;C+) is a sub-

set of *(τ1, τ2;C+) depends on the values τ1, τ2, s− (assuming s+ = s− + 1) and h

and that in most cases *h,GL(τ1, τ2;C+) �⊂ *(τ1, τ2;C

+). Below we give some analy-sis.

When τ2 = s−h (i.e. n2 = s−), the first of the two terms in (4.2) is absentand we consider sets *(τ1, s−h;C

+) and *h,GL(τ1, s−h;C+). Figure 2, where bound-

aries of these sets are shown for τ1 �= 0 and τ1 = 0, suggests that the bound-ary of *h,GL(τ1, τ2;C

+) can closely approximate the boundary of *(τ1, τ2;C+) when

τ1 �= 0. The corresponding results are shown in figure 3 for τ2 > s−h and τ1 �= 0.Here an enlargement of the left parts (i.e. the most illustrative parts) of the bound-aries of *h,GL and * is depicted for different values of τ1, τ2 and s−. If τ1 = 0,then *h,GL(0, τ2;C

+) �⊂ *(0, τ2;C+) for any values of τ2, cf. figure 4. Note that

the size of the external part of *h,GL(0, τ2;C+) with respect to *(0, τ2;C

+) decreaseswith h.

We conclude that the proposed quadrature method assures condition (3.12) forany h (satisfying τ1 = n1h, τ2 = n2h) under the following conditions on the Lagrangeinterpolation: s− � τ1/h and s+ = s− + 1. If τ1 �= 0, the first condition can be satisfied


(a) (b)

Figure 2. Boundaries of *(τ1, s−h;C+) and *h,GL(τ1, s−h;C+) are denoted by solid, respectivelydashed lines. (a) h = 1, τ1 = 1, s− = 2; (b) h = 1, τ1 = 0, s− = 1.

(a) (b) (c)

Figure 3. Parts of the boundaries of *(τ1, τ2;C+) (solid line) and *h,GL(τ1, τ2;C+) (dashed line).(a) h = 1, τ1 = 1, τ2 = 3, s− = 2; (b) h = 1, τ1 = 1, τ2 = 4, s− = 2; (c) h = 1, τ1 = 2,

τ2 = 4, s− = 3.

by choosing h � τ1/s− for a given s−. If τ1 = 0, then (3.12) is only satisfied by us-ing the Lagrange polynomial of degree 1, i.e. s− = 0, s+ = 1 (equivalently, using thetrapezoidal rule).

It is interesting to note that in an LMS-approximation of DDEs coupled with La-grange interpolation for delayed solution terms, the condition s− � s+ � s− + 2 is usedto retain the stability of the original equation (see, e.g., [10,25]).

4.2. Newton–Cotes formulas

For m-point Newton–Cotes formulas with m � 3, numerical experiments showedthat *h,NC(τ1, τ2;C

+) �⊂ *(τ1, τ2;C+) for any values of τ1 and τ2, see figures 5 and 6.

Moreover, when τ1 = 0, the size of the external part of *h,NC(0, τ2;C+) with respect to


(a) (b)

Figure 4. ∂*(0, τ2) and *h,GL(0, τ2;⋃

θ∈Riθ) are denoted by solid, respectively dotted (h = 1) and

dashed (h = 0.5) lines. τ2 = 4; (a) s− = 1; (b) s− = 2.

Figure 5. Solid and dashed lines denote ∂*(τ1, τ2), respectively ∂*h,NC(τ1, τ2), τ1 = 1, τ2 = 4, h = 1,m = 4. Here ∂*h,NC(·, ·) denotes the boundary of *h,NC(·, ·;⋃θ∈R

iθ).

*(0, τ2;C+) does not decrease significantly with h in contrast to the situation with the

proposed quadrature scheme (compare figures 4 and 6). The latter is due to the followingproperty of Newton–Cotes formulas.

Let

Sh(τ1, τ2; θ) :=nm∑r=1

m∑l=1

ωle−iθh(gr−l+1) = e−iθτ1

nm∑r=1

e−iθh(r−1)(m−1)m∑l=1

ωleiθh(l−m),

θ ∈ R. (4.3)

Obviously,⋃

θ∈RSh(τ1, τ2; θ) = *h,NC(τ1, τ2;⋃θ∈R

iθ), where *h,NC is definedby (3.14).


(a) (b) (c)

Figure 6. ∂*(0, hm) and *h,NC(0, hm;⋃θ∈Riθ) are denoted by solid, respectively dotted (h = 1) and

dashed (h = 0.5) lines. hm = (m − 1)h, m = 3, 4, 5. Points Mm are denoted by ◦.

Let Mm := maxθ∗ |�(Sh(0, hm; θ∗))|, where θ∗ is such that �(Sh(0, hm; θ∗)) < 0and �(Sh(0, hm; θ∗)) = 0, see figure 6. We will show that Mm does not depend on h.Let m̃ := �m/2�. Since ωl = ωm−l+1, l = 1, . . . , m̃, we obtain

Sh(0, hm; θ) = eiθh(1−m)/2

(m̃∑l=1

ωl

(eiθh(l−(m+1)/2) + e−iθh(l−(m+1)/2))+ kωm̃+1

)= eiθh(1−m)/2Q,

Q :=m̃∑l=1

2ωl cos

(θh

(l − (m + 1)

2

))+ kωm̃+1 ∈ R,

(4.4)

where k = 0 if m is even and k = 1 otherwise. Hence �(Sh(0, hm; θ∗j )) = 0 for

θ∗j = 2πj/(1 − m)h, j = 1, 2, . . . . We halve the stepsize h and define h̃ := h/2,

h̃m := hm/2 and ω̃l := ωl/2. Using (4.3) and nm = hm/h̃m = 2, we have

Sh(0, hm; θ) = (1 + e−iθh̃(m−1)

)S̃h(0, h̃m; θ), (4.5)

where S̃h(0, h̃m; θ) is defined similar to (4.4) using ω̃l and h̃ instead of ωl , respectively h.Hence �(S̃h(0, h̃m; 2θ∗

j )) = 0 and �(S̃h(0, h̃m; 2θ∗j )) = �(Sh(0, hm; θ∗

j ))/2. It follows

from (4.5) that �(Sh(0, hm; 2θ∗j )) = 2�(S̃h(0, h̃m; 2θ∗

j )). As a consequence, Mm re-mains constant when changing h. Some examples (M3 = −2/3, M4 = −3/8, M5 =−76/45) are depicted in figure 6.

5. Numerical experiments

In this section we comment on the conditions obtained to assure the preserva-tion of RHP-stability of (1.1) by a discretized equation and we present results on theconvergence orders of numerical approximations to the rightmost roots of the character-istic equation for (1.1). We consider DIDEs with constant and nonconstant kernel func-


tions. Note that results similar to theorems 3.1 and 3.2 can be proven for the considered(bounded and analytic) kernel functions.

Characteristic roots of the discretized equation are computed as follows. Once thematrix-approximation of the solution operator to (1.1) over one time step h is constructed(cf. section 2), we compute its eigenvalues. Note that the computational cost is thusdetermined by the size of this n×n matrix, where n ≈ τ2/h. The computed eigenvalues,µ, approximate (a finite number of) the eigenvalues of the solution operator which areexponential transform of the roots λ of the characteristic equation for (1.1). Let λh denotethe characteristic roots for a discretized equation, to avoid confusion with roots λ. Hence,once µ is computed, λh can be extracted using µ = eλ

hh [10]:

�(λh) = 1

hln(|µ|), �(λh

) = 1

harcsin

(�(µ)

|µ|)

modπ

h.

5.1. Preservation of stability

Here we illustrate how a violation of conditions (3.11) and (3.12) may influencethe preservation of RHP-stability of (1.1) by the discretized equation.

Condition (3.11). As an illustrative example, we choose a DIDE with A0 = −100,A1 = −5, K = 1, τ1 = 1 and τ2 = 2. The zero solution of this equation is sta-ble, cf. (3.9). When applying the 3-step Adams–Bashforth method coupled with theproposed quadrature scheme to this equation, condition (3.11) gives h < 0.0052. Nu-merical tests (with condition (3.12) satisfied) showed that the discretized equation isunstable when using h larger than h 0.00545, e.g., the use of h = 0.0055 leads to therightmost characteristic roots λh

1,2 ≈ 1.38 ± 5701.2i. Hence this example illustrates thatcondition (3.11) is not necessarily over-restrictive and can be quite close to the conditionfor real stability of the approximation in a number of cases.

As a side remark, note that if the equation with A0 = −100, A1 = −5 is coupledto a similar equation with small |A0|, |A1| (e.g., A0 = −2, A1 = 1), one could speak of astiff example where the steplength is restricted by stability (through the large |A0|, |A1|)more than by accuracy (as determined by the small |A0|, |A1|). Because we considerhere only the scalar case, we leave a more detailed discussion of this issue for futureresearch.

For equation (1.1) with a constant kernel, a common approach is to remove theintegral term by differentiation. The original equation is strictly speaking not completelyequivalent to any of the following equations:

d2

dt2y(t) = A0

d

dty(t) + A1

(y(t − τ1) − y(t − τ2)

)(5.1a)


ord

dty1(t) = A0y1(t) + A1y2(t),

d

dty2(t) = y1(t − τ1) − y1(t − τ2),

(5.2)

since (i) the zero solution is the only steady state solution of (1.1) while any constantsolution satisfies (5.1a) and a set of steady state solutions, (y∗

1 , y∗2 ) = (y∗

1 ,−A0y∗1/A1),

y∗1 ∈ R, satisfies (5.2); (ii) the stability of the zero solution of (1.1) is not equivalent to

the stability of the zero solution of (5.1a) and (5.2) since the characteristic equation forthe latter always has a zero root (additional comments can be found in [5]). Nevertheless,for practical purposes, one can always investigate stability of (5.1a) or (5.2) insteadof (1.1) as all characteristic roots, except for the extra characteristic root λ = 0, are thesame as for (1.1). Note that the condition on h, obtained in [10] to preserve stabilityof a DDE by an LMS-approximation combined with Lagrange interpolation for delayedterms, when applied to (5.1a) and (5.2) with A0, A1, τ1, τ2 as in the example consideredabove gives the same results as the ones determined by (3.11) for the DIDE.

Condition (3.12). In order to analyze consequences of violation of condition (3.12),we use the trapezium rule as an LMS method. It is an A-stable rule and hence condi-tion (3.10a) is fulfilled for all h. If *ch(C

+) ⊆ *c(C+) (equivalently, *h(τ1, τ2;C

+)⊆ *(τ1, τ2;C

+)) then RHP-stability is recovered by the discretized equation.We showed that *h(τ1, τ2;C

+) �⊂ *(τ1, τ2;C+) for any τ1 and τ2 when using

Newton–Cotes formulas. As an illustrative example, we use (1.1) with K = 1 and A0,A1, τ1 and τ2 such that *c(C

+) ⊂ C−0 , cf. figure 7, and hence the zero solution of (1.1)

is stable. The use of Simpson’s rule with h = τ1 for this example gives the rightmostcharacteristic roots λh

1,2 0.396 ± 49.737i, although the outside part of the region

(a) (b)

Figure 7. (a) Regions *c(C+) and *ch(C

+) are bounded by a solid, respectively dashed line. A0 = −186,A1 = −4905, K = 1, τ1 = 0.0285 and τ2 = 0.0855. *ch corresponds to Simpson’s rule; (b) a blow up of

the left figure.


*ch(C+) with respect to *c(C

+) is relatively small. Decreasing h, the boundary of *ch

approaches the boundary of *c and the numerical solution becomes stable. Note that theuse of the proposed quadrature scheme gives stable roots for any h = τ1/n, n = 1, 2, . . .(here τ2 = 3τ1).

When using the proposed quadrature method in case of a constant kernel, numericalexperiments have shown that if condition (3.12) is violated then the discretized equationmay have roots with (very) small positive real part only if the DIDE has roots with (very)small negative real part, i.e. the DIDE is close to a Hopf bifurcation.

All our tests with nonconstant kernel functions (combinations of trigonometricand polynomial functions) have shown that *h,GL(τ1, τ2;C

+) ⊆ *(τ1, τ2;C+) in case

τ1 �= 0. However there is no guarantee that this property holds in general. Neverthe-less, we expect that when the property is violated, the approximation of the boundary of*(τ1, τ2;C

+) by the boundary of *h,GL(τ1, τ2;C+) should be good enough to allow a

correct approximation of the rightmost characteristic roots.Experiments with τ1 = 0 have suggested to distinguish between two types of kernel

functions: in a neighborhood of ξ = 0, as ξ → 0, (a) K(ξ) is a decreasing function and(b) K(ξ) is not a decreasing function. In applications these types of K(ξ) can correspondto situations when: (a) the effect of a certain moment in the past, e.g. at time t − τ , issmoothed over a finite interval [t − τ2, t] such that K(ξ) peaks at ξ = τ , 0 < τ < τ2,and is small elsewhere, and (b) an infinite distributed delay is approximated by a finiteone, cutting off a decaying (as ξ → ∞) kernel function.

Some results of numerical experiments corresponding to the case (a) are givenin table 1 for K(ξ) = sin(ξ) + α. Numerically obtained estimates of h presented inthis table guarantee that *h,GL(0, τ2;C

+) ⊂ *(0, τ2;C+). Clearly, these estimates are

sufficient conditions and the corresponding discretized equation may retain stability ofa DIDE for larger h. For instance, the discretized equation for (1.1) with A0 = −3.5,A1 = −1, τ2 = 3, α = 5 is unstable, in case s− = 1, for h = τ2, τ2/2 (λh

1,2 ≈0.35 ± 2.09 i for h = τ2/2) and is stable for h � τ2/3: λh

1,2 ≈ −0.07 ± 1.69 i forh = τ2/3. The rightmost roots of (3.1) in this case are λ1,2 ≈ −0.014 ± 1.741i.

In the case (b), all our experiments have shown that *h,GL(0, τ2;C+) �⊂

*(0, τ2;C+) for any h. This case includes a constant kernel function. Hence, results

Table 1Numerically obtained estimates for the step size h (h = τ2/n, n = 1, 2, . . .) which guarantee that

*h,GL(0, τ2;C+) ⊂ *(0, τ2;C+). Here K(ξ) = sin(ξ) + α, ξ ∈ [0, τ2].τ2 = 1.5 τ2 = 3

α = 0 α = 1 α = 5 α = 0 α = 1 α = 5

s− = 0 h � τ2 h � τ2 h � τ2 h � τ2 h � τ2 h � τ2

s− = 1 h � τ2

2h � τ2

3h � τ2

6h � τ2

3h � τ2

4h � τ2

10

s− = 2 h � τ2

2h � τ2

4h � τ2

10h � τ2

3h � τ2

5h � τ2

18


shown in figure 4 and the above consideration for a constant kernel are rather illustrativefor nonconstant kernels of this type: the boundary of *h,GL approaches the boundary of* as h → 0 and the discretized equation can have roots with very small positive realpart only if the DIDE is close to a Hopf bifurcation.

5.2. Results on the convergence orders for the characteristic root approximations

Our tests on the convergence orders for the characteristic root approximations giveresults similar to the proven results on the convergence order for solutions of numericalschemes for DIDEs obtained by applying an LMS method combined with a quadra-ture rule [4]. Namely, the O(hmin(k,q+1)) convergence is clearly apparent when using ak-step LMS method and a quadrature rule with the degree of precision q, cf. table 2. Inthe presented tests, we used the implicit backward differentiation (BDF) method withdifferent number of steps, k = 3, 4, 5, 6, and the proposed quadrature method withm = s−+1, s− = 1, 2, i.e. the quadrature has degree of precision, q = 2m−1 = s−+s+,equal to the degree of the Lagrange polynomial that we use (assuming s+ = s− + 1).

Table 2Numerically observed orders of convergence while approximating the rightmost root λ1 of (3.4) (example 1)and the second root λ2 of (3.1) (example 2) using BDF methods with k = 3, 4, 5, 6 steps and Gauss–Legendre quadratures with m = 2, 3 (i.e. s− = 1, 2). Example 1: A0 = −4, A1 = −3, τ1 = 1, τ2 = 4.

Example 2: A0 = −3, A1 = 2, τ1 = 2, τ2 = 5, K(ξ) = (5 − ξ) cos(6ξ) + 2.5, ξ ∈ [2, 5].Example 1 (constant kernel) Example 2 (nonconstant kernel)

k = 3 k = 4 k = 5 k = 6 k = 3 k = 4 k = 5 k = 6

m = 2 2.990 3.978 4.083 4.005 3.010 4.024 3.825 3.979m = 3 2.997 3.996 5.007 5.997 3.000 4.000 5.002 6.004

Figure 8. For example 1, convergence of the rightmost characteristic root of (3.5), λh1, to the root λ1 forvarying h = 1/L, L = 10, . . . , 100, and different number of steps k of BDF methods, k = 3 (×), k = 4 (◦),

k = 5 (+) and k = 6 (∗). Here Gauss–Legendre quadrature was used with m = 3 (i.e. s− = 2).


(a) (b)

Figure 9. The computed rightmost characteristic roots for example 1, τ1 = 1 (×), τ1 = 0 (◦) (a) and forexample 2 (b).

The above estimates were obtained as follows. We computed characteristic rootscorresponding to the two examples of DIDEs, cf. the caption of table 2, for varying h,h = 1/L, L = 10, . . . , 100. Taking the rightmost root λh

1 in example 1 and the sec-ond root, λh

2, in example 2, we estimated, based on a least squares approximation of theobtained results, the order of convergence of these roots towards the “exact” roots λ1, re-spectively λ2. The “exact” roots, λ1 0.069− 1.176 i and λ2 −0.021− 1.649 i, werecomputed with accuracy 10−13 using a Newton iteration on the corresponding character-istic equation. Figure 8 illustrates convergence of the root λh

1 to the root λ1 for example 1.Note that the rates of convergence of all roots are similar but have different error con-stants. The rightmost characteristic roots for example 1 (including case τ1 = 0) and forexample 2 are depicted in figure 9.

6. Conclusion

Numerical stability analysis and computation of characteristic roots of delayintegro-differential equations (DIDEs) is still an open area. In practical computations,one should use numerical schemes that retain the stability properties of the original equa-tion. The analysis presented in this paper addresses this problem for scalar DIDEs withcommensurate delays.

We have proposed a numerical scheme for computing characteristic roots of DIDEswhich involves a linear multistep method as time integration scheme and a quadraturemethod based on Lagrange interpolation and a Gauss–Legendre quadrature rule. Aimingat the analysis of stability properties of the proposed scheme, we derived and proved asufficient condition for the (asymptotic) stability of a DIDE with a constant kernel whichwe call RHP-stability. We obtained conditions under which the stability is preservedby the proposed scheme. When these conditions are not fulfilled, we showed that the


proposed scheme remains preferable to Newton–Cotes formulas which do not retainRHP-stability.

Using the proposed scheme, we present results on the convergence orders of nu-merical approximations to the rightmost (stability determining) roots of the characteris-tic equation associated with the original equation. These results are in agreement withproven results on the convergence order for solutions of numerical schemes for DIDEsobtained by applying an LMS method combined with a quadrature rule [4].

One of the most important issues in the stability analysis of DIDEs is the effectof the kernel function on the stability of the applied numerical scheme. Our analyticalresults for a constant kernel and numerical experiments for nonconstant kernels suggestthat the proposed scheme can retain RHP-stability of the original equation with a rathergeneral kernel if the effect of the past is distributed over the interval [t − τ2, t − τ1] withτ2 > τ1 > 0, i.e. τ1 �= 0.

Future research can be directed towards systems of DIDEs with non-commensuratedelays and when the delay τ1 equals or approaches zero.

Acknowledgements

This research presents results of the Research Project OT 98/16, funded by the Re-search Council K.U. Leuven, of the Project G.0270.00 funded by the Fund for ScientificResearch – Flanders (Belgium) and of the Project IUAP P5/22 funded by the programmeon Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’sOffice for Science, Technology and Culture. K. Engelborghs is a Postdoctoral Fellowof the Fund for Scientific Research – Flanders (Belgium). We thank the anonymousreferees for their helpful comments which improved the presentation of the paper.

Appendix A. Proofs of theorems

A.1. Necessary results

To prove theorems 3.1–3.3, we need the following lemma.

Lemma A.1. If f (z) is analytic and lim|z|→∞ f (z) = z∞ ∈ C uniformly then⋃�(z)�0 f (z) is inside the curve

⋃θ∈R

f (iθ).

Proof. Let M > 0 be such that |f (iθ)| < M, ∀θ ∈ R. Let a point q ∈ C be outside theregion bounded by

⋃θ∈R

f (iθ). Then there exists a curve C0 starting at q and ending ata point b with |b| > M which does not intersect

⋃θ∈R

f (iθ).Consider the contour CR which consists of the imaginary axis up to {iθ | |θ | � R}

and the semicircle {z | �(z) � 0, |z| = R}. Let r0 denote the smallest distance ofthe curve C0 to the point z∞. We choose R big enough such that |f (z) − z∞| < r0 for|z| � R. Let the closed contours JR and QR define the images of the contour CR under


the mappings J = f (z), respectively Q = fq(z), fq(z) := f (z) − q. It follows fromthe above that the curve C0 and the contour JR (equivalently, curves C0 − q and QR)do not intersect. As a consequence, q lies outside JR, and, equivalently, the origin liesoutside QR.

This implies that the curve QR does not encircles the origin, and, due to the argu-ment principle applied to fq(z) on the closed contour CR, the number of zeros of fq(z)

equals the number of poles of fq(z) in the interior of CR. Since fq(z) has no poles, itcan have no zeros and hence f (z) �= q within CR. As the latter holds for R arbitrarilylarge, it is clear that f (z) �= q for all z, �(z) � 0 which proves the lemma. �

To prove theorem 4.1, we need certain properties of coefficients ωl of the quadra-ture (2.4),

ωl =m∑

k=1

BkPl(εk), l = −s−, . . . , s+. (A.1)

Recall that εkh and Bk are the zeros of the mth-degree Legendre polynomial, respec-tively weights of the Gauss–Legendre quadrature on [0, h] and Pl(εk) are the Lagrangecoefficients,

Pl(εk) =s+∏

ν=−s−, ν �=l

εk − ν

l − ν, εk ∈ [0, 1], l = −s−, . . . , s+. (A.2)

Let v(ε) := (ε + s−)(ε + s− − 1) · · · (ε + 1)ε(ε − 1) · · · (ε − s+) and v′(j) =(dv(ε)/dε)ε=j . Then (A.2) reads

Pl(εk) = v(εk)

(εk − l)v′(l) . (A.3)

Assume that s+ = s− + 1. Points εk and εm−k+1, k = 1, . . . , �m/2� (�r� de-notes the smallest integer less than or equal to r ∈ R), are symmetric with respect to1/2, i.e. εk = 1 − εm−k+1. Note also that v(εk) = v(1 − εk) and v′(l) = −v′(1 − l),l = −s−, . . . , 0. Hence, using (A.3), we obtain

Pl(εk) = P1−l(1 − εk), Pl(0.5) = P1−l(0.5), k = 1, . . . ,

⌊m

2

⌋, l = −s−, . . . , 0.

(A.4)Using (A.4) and that Bk = Bm−k+1, k = 1, . . . , �m/2�, [23], we obtain

ωl =m∑

k=1

BkPl(εk) =m∑

k=1

BkP1−l(1 − εk) =m∑

k=1

BkP1−l(1 − εm−k+1)

=m∑

k=1

BkP1−l(εk) = ω1−l, l = −s−, . . . , 0. (A.5)


A.2. Proof of theorem 3.1

Let F(λ) := ∫ τ2τ1

e−λξ dξ . F(λ) is analytic and F(λ) → 0 uniformly as |λ| → ∞.Hence, due to lemma A.1,

⋃�(λ)�0 F(λ) is inside the curve

⋃θ∈R

F(iθ). �


Let F(λ) := ∑ncr=1

∑s+l=−s− ωle−λh(q1,r−l) +∑ns

r=1

∑s(r)+l=−s(r)− ωl,re−λh(q2,r−l).

If τ1 � (s+ − 1)h, then nc = n and ns = 0. Hence

F(λ) =n∑

r=1

s+∑l=−s−

ωle−λh(n2−r+1−l)

and lim|λ|→∞ F(λ) = ωs+ uniformly if τ1 = (s+ − 1)h, and lim|λ|→∞ F(λ) = 0 uni-formly, otherwise.

If τ1 < (s+ − 1)h, then nc = n2 − s+ + 1 and ns = s+ − 1 − n1. Hence

F(λ) =n2−s++1∑

r=1

s+∑l=−s−

ωle−λh(n2−r+1−l) +

s+−1−n1∑r=1

s(r)+∑l=−s(r)−

ωl,re−λh(s+−r−l)

and lim|λ|→∞ F(λ) = ∑s+−1−n1r=1 ωs(r)+,r uniformly if τ2 = (s+ − 1)h, and

lim|λ|→∞ F(λ) = ωs+ +∑s+−1−n1r=1 ωs(r)+,r uniformly, otherwise.

Due to lemma A.1,⋃

�(λ)�0 F(λ) is inside the curve⋃

θ∈RF(iθ). �


Let F(λ) := ∑nmr=1

∑ml=1 ωle−λh(qr−l+1) = e−λτ1

∑nmr=1 e−λh(r−1)(m−1) ×∑m

l=1 ωleλh(l−m). If τ1 = 0, then lim|λ|→∞ F(λ) = ωm uniformly and lim|λ|→∞ F(λ)

= 0 uniformly otherwise. Hence, due to lemma A.1,⋃

�(λ)�0 F(λ) is inside thecurve

⋃θ∈R

F(iθ). �


Let

S(θ) :=∫ h

0e−iθξdξ = sin(θh) + i(cos(θh) − 1)

θ,

Sh(θ) :=s+∑

l=−s−

ωleiθh(l−1), θ ∈ R,

∂*(0, h) :=⋃

θ∈[−2π/h,2π/h]S(θ), ∂*h,GL(0, h) :=

⋃θ∈[−π/h,π/h]

Sh(θ).


Obviously,

*

(0, h;

⋃θ∈R

iθ

)≡⋃θ∈R

S(θ), *h,GL

(0, h;

⋃θ∈R

iθ

)≡⋃θ∈R

Sh(θ).

The proof of the theorem consists of the following parts which we prove succes-sively below.

(i) *(0, h;⋃θ∈Riθ) is bounded by the closed curve ∂*(0, h). *h,GL(0, h;⋃θ∈R

iθ)is equivalent to the closed curve ∂*h,GL(0, h).

(ii) The curve ∂*h,GL(0, h) is inside the curve ∂*(0, h).

(iii) The curve *h,GL(τ1, τ2;⋃θ∈Riθ) is inside the curve *(τ1, τ2;⋃θ∈R

iθ).

(iv) The result of the theorem follows from theorems 3.1, 3.2 and (iii).

(i) Let

S0(θ) := S(θ), θ ∈[−2π

h,

2π

h

],

S+j (θ) := S(θ), θ ∈

[2πj

h,

2π(j + 1)

h

],

S−j (θ) := S(θ), θ ∈

[−2πj

h,−2π(j + 1)

h

], j = 1, 2, . . . .

The curve⋃

θ∈RS(θ) can be considered as an union of branches S0(θ) and S±

j (θ), j =1, 2, . . . . All these branches are closed curves since S(θ) is symmetric w.r.t. the realaxis and �(S(0)) = �(S(±2πj/h)) = �(S(±2πj/h)) = 0, j = 1, 2, . . . . Con-sidering S(θ) as a 2π/h periodic function in θ divided by θ , it follows that |S+

j | �|S0|, arg(S+

j (θ̃ )) = arg(S0(θ)), θ̃ ∈ [2πj/h, 2π(j + 1)/h], θ ∈ [0, 2π/h], and sim-ilar for S−

j , j = 1, 2, . . . . Therefore S±j , j = 1, 2, . . ., are located inside S0, i.e. in-

side ∂*.The second part is due to: Sh(θ) is 2π/h periodic, symmetric with respect to the

real axis and �(Sh(0)) = �(Sh(±π/h)) = 0.(ii) Now we write S and Sh as

S(θ) = h

∫ 1

0eiθh(ε−1)dε = sin(θh) + i(cos(θh) − 1)

θ, ε ∈ [0, 1],

Sh(θ) = h

∫ 1

0P(iθh, ε)dε =

s+∑l=−s−

ωleiθh(l−1), (A.6)

P(iθh, ε):=s+∑

l=−s−

Pl(ε)eiθh(l−1).

The result follows from the following two steps which we will prove below:


(1) arg(Sh(θ)) = arg(S(θ)), θ ∈ [−π/h, π/h];(2) |Sh(θ)| � |S(θ)|, θ ∈ [−π/h, π/h].

Step 1. Using the assumption s+ = s− + 1 and (A.5), we obtain

Sh(θ)=0∑

l=−s−

ωl

(eiθh(l−1) + eiθh(−l)

) = e−iθh/20∑

l=−s−

ωl

(eiθh(l−1/2) + e−iθh(l−1/2))

= e−iθh/20∑

l=−s−

2ωl cos

(θh

(l − 1

2

))= e−iθh/2Q, Q ∈ R, θ ∈

[−π

h,π

h

].

(A.7)

Hence arg(Sh(θ)) = −θh/2. It follows from (A.6) that

arg(S(θ)

) = tan−1

(cos(θh) − 1

sin(θh)

)= tan−1

(− sin(θh/2)

cos(θh/2)

)= −θh

2, θ ∈

[−π

h,π

h

].

Step 2. Consider functions P(iθh, ε) and eiθh(ε−1), i.e. integrands in the definitionof Sh(θ) and S(θ), cf. (A.6). Let z := eiθh. Then P(iθh, ε) can be considered as apolynomial in z. Since |z| = 1, s+ = s− + 1 and |eiθh(ε−1)| = 1, it follows [17,22] forany ε ∈ [0, 1] that

∣∣P(iθh, ε)∣∣ = ∣∣∣∣∣z−s−−1

s+∑l=−s−

Pl(ε)zl+s−

∣∣∣∣∣ � 1, and hence∣∣P(iθh, ε)

∣∣ �∣∣eiθh(ε−1)

∣∣.(A.8)

Now consider P(iθh, ε) and eiθh(ε−1) as functions in ε (cf. figure 10). For any θ ∈(0, π/h) we have:

(a) P(iθh, ε)|ε=0 = eiθh(ε−1)|ε=0 = e−iθh, P(iθh, ε)|ε=1 = eiθh(ε−1)|ε=1 = 1.

(b) �(eiθh(ε−1)) < 0 and �(P (iθh, ε)) < 0, ε ∈ (0, 1).

The latter follows from the following. For s− = 0 we have P(iθh, ε) = P0(ε)e−iθh +P1(ε) = (1− ε)e−iθh + ε and hence �(P (iθh, ε)) < 0. For a fixed s−, let Q1 and Q2 de-note P0(ε) associated with s−−1 and s−, respectively. Then Q2/Q1 = (ε+s−)(s+−ε)/

(s−s+) 1. Hence P0(ε) does not depend on s− significantly and similar holds forPl, l = −s−, . . . , s+. Since

∑s+l=−s− Pl(ε) = 1, it follows that coefficients P0(ε) and

P1(ε) are dominant ones. As a result, �(P (iθh, ε)) < 0, ∀s−.Note that property (b) implies that P(iθh, ε) does not cross the real axis whenever

ε ∈ (0, 1) and θ ∈ (0, π/h). Due to (A.8), (a) and (b), we conclude that |Sh(θ)| <

|S(θ)| for θ ∈ (0, π/h). Taking into account that |Sh(0)| = |S(0)| = h, |Sh(π/h)| = 0and |S(π/h)| = 2h/π , it follows that |Sh(θ)| � |S(θ)|, θ ∈ [0, π/h].

Due to the symmetry of Sh(θ) and S(θ) w.r.t. the real axis, |Sh(θ)| < |S(θ)|, θ ∈[−π/h, 0).


Figure 10. In the complex plane, P(iθh, ε) (dashed curve) and eiθh(ε−1) (solid curve) as functions in ε,ε ∈ [0, 1]. h = 1, θ = 2.5, s− = 1, s+ = 2.

(iii) The result is due to (ii) and that

*

(τ1, τ2;

⋃θ∈R

iθ

)=⋃θ∈R

(∫ τ2

τ1

e−iθξ dξ

)=⋃θ∈R

(e−iθτ1

n∑r=1

e−iθ(r−1)h∫ h

0e−iθξ dξ

)

=⋃θ∈R

(Q(θ)

∫ h

0e−iθξ dξ

), Q(θ) := e−iθτ1

n∑r=1

e−iθ(r−1)h ∈ C,

*h,GL

(τ1, τ2;

⋃θ∈R

iθ

)=⋃θ∈R

(e−iθτ1

n∑r=1

e−iθ(r−1)hs+∑

l=−s−

ωleiθh(l−1)

)

=⋃θ∈R

(Q(θ)

s+∑l=−s−

ωleiθh(l−1)

). �

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Computing Stability of Differential Equations with Bounded Distributed Delays

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