A Bibliography on the Numerical Solution of Delay Differential Equations.pdf

A Bibliography on the Numerical Solution of
Delay Di erential Equations
C.T.H. Baker, C.A.H.Paul & D.R. Will
e
Numerical Analysis Report No. 269
Version 1.0
June 1995
University of Manchester/UMIST
Manchester Centre for Computational Mathematics
Numerical Analysis Reports
DEPARTMENT OF MATHEMATICS
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University of Manchester
Manchester M13 9PL
England
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A Bibliography on the Numerical Solution of
Delay Di erential Equations
C.T.H. Baker
, C.A.H. Paul
and D.R. Will
ey
June 15, 1995 | last updated June 15, 1995
Abstract
The aim of this bibliography is to provide an introduction to papers and technical reports in the eld of
delay di erential equations and related di erential equations. In addition to the title, authors and reference of
an article, we provide the abstract which, if the article has previously appeared as a technical report, comes from
the published paper unless indicated by a z. The main interest in this bibliography derives from the references
to early papers and technical reports in the eld, as nowadays on-line search facilities (such as BIDS in the
U.K.) provide access to the most recent publications. Although it is hoped to keep the bibliography up-to-
date, most immediate e ort will be invested in extending the collection of earlier references { as it is far from
being exhaustive. The up-to-date bibliography is only available by anonymous ftp, since it is hoped to update it
regularly.
Key words. Delay di erential equations, numerical solution, stability, applications.
AMS subject classi cations. 65Q05
Contents
1 Dense-Output Methods 2
2 Delay Di erential Equations 8
2.1 Numerical Methods : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8
2.2 Dynamics and (Numerical) Stability : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19
2.3 Applications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30
3 Functional Di erential Equations 37
3.1 Numerical Methods : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 37
3.2 Dynamics and (Numerical) Stability : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 45
3.3 Applications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 47
Introduction
The main aim of this bibliography is to provide references to papers and technical reports on most aspects of delay
di erential equations (DDEs). Currently, most DDE solvers are constructed using adapted continuous numerical
methods for solving ordinary di erential equations (ODEs). Thus, this bibliography starts with a short section on
dense-output methods for ODEs. Additionally, there is an Author Index at the end of this report to facilitate
nding speci c papers/reports.
Mathematics Department, The Victoria University of Manchester, Manchester M13 9PL, England.
yMathematical Applications, Ciba-Geigy AG, Basel, Switzerland.
1

1 Dense-Output Methods
Title: Stability Properties of Interpolants for Runge-Kutta Methods
Authors: A Bellen and M Zennaro
Reference: SIAM J. Numer. Anal., 25 (1988), 411{432.
Abstract: In solving initial value problems for ordinary di erential equations by means of a discrete method, we
often need to know the solution on a set of points that di ers from the grid (systems with driving equation, dense
output, discontinuity problems, delay equations, etc.). The non-nodal approximations are generally obtained by
local interpolation, and di erent features can be requested from the interpolants according to the problem treated.
In this paper we introduce the concept of stable interpolants for Runge-Kutta methods. Roughly speaking, a stable
interpolant maintains the stability properties of the Runge-Kutta method itself. The lack of such stability makes
the continuous extension not reliable, especially in solving sti equations.
Title: Interpolating High-order Runge-Kutta formulae
Authors: P Bogacki and L F Shampine
Reference: Comp. Math. Applic., 20(3) (1990), 15{24.
Abstract: A Runge-Kutta formulabecomes inecient when the stepsize must be reduced often to produce answers
at speci ed points. Recently, a lot of e ort has been devoted to providing Runge-Kutta formulae with an inter-
polation capability so that answers can be produced inexpensively throughout the span of a step. Though quite
successful at low and moderate orders, interpolation of high-order formulae is still unsatisfactory. A new approach
to the task is presented in the context of interpolating an important pair of orders 7 and 8 due to Dormand and
Prince.
Title: Collocation at Gaussian Points
Authors: C de Boor and B Swartz
Abstract: Approximations to an isolated solution of an m-th order nonlinear ordinary di erential equation with
m linear side conditions are determined. These approximations are piecewise polynomial functions of order m +k
(degree less than m +k) possessing m ?1 continuous derivatives. They are determined by collocation, i.e., by the
requirement that they satisfy the di erential equation at k points in each subinterval, together with the m side
conditions. If the solution of the suciently smooth di erential equation problem has m+2k continuous derivatives
and if the collocation points are the zeros of the k-th Legendre polynomial relative to each subinterval, then the
global error in these approximationsis O(jjm+k) with jj the maximumsubinterval length. Moreover, at the ends
of each subinterval, the approximation and its rst m ? 1 derivatives are O(jj2k) accurate. The solution of the
nonlinear collocation equations may itself be approximated by solving the sequence of linear collocation problems
associated with a Newton iteration; convergence of this process is locally quadratic.
Title: An Implementation of Singly-Implicit Runge-Kutta Methods
Authors: K Burrage, J C Butcher and F H Chipman
Reference: BIT, 20 (1980), 326{340.
Abstract: A description is given of STRIDE, an algorithm for the numerical integration of ordinary di erential
equations. This algorithm, which is applicable to either sti or non-sti initial value problems, is based on the
family of singly-implicit Runge-Kutta methods of Burrage. The present paper is con ned mainly to a theoretical
discussion, but includes an overview of the structure of the algorithm together with a general description of how it
is used. A companion report contains more detailed documentation intended particularly for a potential user of the
algorithm. The companion report also includes an Algol 60 procedure declaration for STRIDE together with the
listing of an equivalent Fortran subroutine.
2

Title: A Transformed Implicit Runge-Kutta Method
Authors: J C Butcher
Reference: J. Assoc. Comput. Mach., 26 (1979), 731{738.
Abstract: Certain implicit Runge-Kutta methods are capable of being transformed into a form which makes the
modi ed Newton iterates in their implementation capable of ecient computation. For the class of such methods
considered, the transformations are given explicitly, and it is shown how error estimates, as well as initial iterates
for a succeeding step, can be expressed in terms of the transformed variables.
Title: A Fifth-Order Interpolant for the Dormand and Prince Runge-Kutta method
Authors: M Calvo, J I Montijano and L Randez
Reference: J. Comp. Appl. Math., 29 (1990), 91{100.
Abstract: A family of fth-order interpolants for the fth-order solution provided by the Dormand and Prince
Runge-Kutta pair RK5(4)7M which requires two additional function evaluations per step is presented. An optimal
interpolant in this family has been determined choosing the available parameters of this family so as to minimize
the leading coecients of the local truncation error of the continuous solution. Finally some numerical experiments
with the non-sti DETEST problems show that the proposed optimal method has a good interpolatory behaviour.
Title: A New Embedded Pair of Runge-Kutta formulae of Orders 5 and 6
Authors: M Calvo, J I Montijano and L Randez
Reference: Comp. Math. Applic., 20(1) (1990), 15{24.
Abstract: A new pair of embedded Runge-Kutta (RK) formulae of orders 5 and 6 is presented. It is derived
from a family of RK methods depending on eight parameters by using certain measures of accuracy and stability.
Numerical tests comparing its eciency to other formulae of the same order in current use are presented. With an
extra function evaluation per step, a C1-continuous interpolant of order 5 can be obtained.
Title: Polynomial and Rational Interpolation in the Numerical Solution of Sti Systems
Authors: S D Colquitt and J Williams
Reference: J. Appl. Math. Comp., 31 (1989), 270{287.
Abstract: We analyze the accuracy of the polynomial interpolation scheme used in standard variable order,
variable step BDF codes for sti systems. In particular the strategy used in the NAG code is considered and shown
to be reliable and robust on a large number of test problems. For the case of monotonic solution components, a
comparison is made with a particularly simple interpolation scheme based on a rational quadratic.
Title: E ective Solution of Discontinuous IVPs using a Runge-Kutta Formula Pair with Interpolants
Authors: W H Enright, K R Jackson, S P Nrsett and P G Thomsen
Reference: Appl. Math. Comp., 27 (1988), 313{335.
Reference: Tech. Rep. 113 (1986), Dept. Math., Univ. Manchester.
Abstractz: An automatic technique for solving discontinuous initial value problems is developed and justi ed.
The technique is based on the use of local interpolants such as those that have been developed for use with Runge-
Kutta formula pairs. Numerical examples are presented to illustrate the signi cant improvement in eciency and
reliability that can be realized with this technique.
Title: Interpolants for Runge-Kutta Formulae
Authors: W H Enright, K R Jackson, S P Nrsett and P G Thomsen
Reference: Appl. Math. Comp., 27 (1988), 313{335.
Abstract: A general procedure for the construction ofinterpolants for Runge-Kutta (RK)formulaeis presented. As
illustrations, this approach is used to develop interpolants for three explicit RK formulae, including those employed
3

in the well-known subroutines RKF45 and DVERK. A typical result is that no extra function evaluations are
required to obtain an interpolant with O(h5) local truncation error for the fth-order RK formula used in RKF45;
two extra function evaluations per step are required to obtain an interpolant with O(h6) local truncation error for
this RK formula.
Title: Analysis of Error Control Strategies for Continuous Runge-Kutta Methods
Authors: W H Enright
Abstract: There has been considerable recent progress in the analysis and the development of interpolation
schemes that can be associated with discrete Runge-Kutta methods. With the availability of these schemes, it can
now be asked that a numerical method provide a continuous approximationto the solution. This paper, rather than
view such a continuous method as an interpolant superimposed on a standard discrete method, consider how the
interpolant and its associated defect can be e ectively used in the underlying error and stepsize control mechanism.
In particular for error control strategies are considered that can be used in methods based on Runge-Kutta formula
pairs with interpolants. An asymptotic and a non-asymptotic analysis of each strategy are presented. It is shown
that a strategy based on direct defect control can provide signi cant advantages over existing strategies with only
a modest increase in cost.
Title: Practical Aspects of Interpolation in Runge-Kutta Codes
Authors: I Gladwell, L F Shampine, L S Baca and R W Brankin
Reference: SIAM J. Sci. Stat. Comp., 8 (1987), 322{341.
Abstract: Runge-Kutta codes solve numerically the initial value problem for a system of ordinary di erential
equations in a step-by-step manner. The automatically chosen step sizes may vary widely in size depending on
the equations, the nature of the error control, and the accuracy desired. Interpolation schemes approximate the
solution of the di erential equation within the step. Analytical tools are developed and applied to a selection of
interpolation schemes with the aim of understanding how the interpolants behave. A novel matter in this context
is the preservation of properties inherent in the values and derivatives computed by the Runge-Kutta code. In
particular, we show how to preserve local monotonicity so as to achieve a visually pleasing interpolant, and in a
way which is especially appropriate to Runge-Kutta codes.
Title: Highly Continuous Runge-Kutta Interpolants
Authors: D J Higham
Reference: ACM Trans. Math. Soft., 17 (1991), 368{386.
Reference: Tech. Rep. 220/89 (1989), Dept. Comp. Sci., Univ. Toronto.
Abstract: To augment the discrete Runge-Kutta solution to the initial value problem, piecewise Hermite inter-
polants have been used to provide a continuous approximation with a continuous rst derivative. We show that it
is possible to construct interpolants with arbitrarily many continuous derivatives which have the same asymptotic
accuracy and basic cost as the Hermite interpolants. We also show that the usual truncation coecient analysis
can be applied to these new interpolants, allowing their accuracy to be examined in more detail. As an illustration,
we present some globally C2 interpolants for use with a popular 4th and 5th order Runge-Kutta pair of Dormand
and Prince, and we compare them theoretically and numerically with existing interpolants.
Title: Fourth- and Fifth-Order, Scaled Runge-Kutta Algorithms for Treating Dense Output
Authors: M K Horn
Abstract: New Runge-Kutta algorithms are developed which determine the solution of a system of ordinary
di erential equations at any point within a given integration step, as well as at the end of each step. The new scaled
4

Runge-Kutta methods are designed to be used with existing Runge-Kutta formulae,using the derivative evaluations
of these de ning algorithms as the core of the new system. Thus, for only a slight increase in computing time,
the solution may be generated within the integration step, improving the eciency of the Runge-Kutta algorithms,
since the step length need no longer be reduced to coincide with the desired output point. Scaled Runge-Kutta
algorithms are presented for orders 4 and 5, along with accuracy comparisons between the de ning algorithms and
their scaled versions for a test problem.
Title: Perturbed Collocation and Runge-Kutta Methods
Authors: S P Nrsett and G Wanner
Reference: Numer. Math., 38 (1981), 193{208.
Abstract: It is well-known that some implicit Runge-Kutta methods are equivalent to collocation methods. This
fact permits very short and natural proofs of order and A, B, AN and BN-stability properties for this subclass
of methods. The present paper answers the natural question, if all RK methods can be considered as a somewhat
perturbed collocation. After having introduced this notion we give a proof on the order of the method and discuss
their stability properties. Much of known theory becomes simple and beautiful.
Title: Order Barriers for Continuous Explicit Runge-Kutta methods
Authors: B Owren and M Zennaro
Reference: Math. Comp., 56 (1991), 645{661.
Abstract: In this paper we deal with continuous numerical methods for solving initial value problems for ordinary
di erential equations, the need for which occurs frequently in applications. Whereas most of the commonly used
multistep methods provide continuous extensions by means of an interpolant which is available without making
extra function evaluations, this is not always the case for one-step methods. We consider the class of explicit
Runge-Kutta methods and provide theorems used to obtain lower bounds for the number of stages required to
construct methods of a given uniform order p. These bounds are similar to the Butcher barriers known for the
discrete case, and are derived up to order p = 5. As far as we know, the examples we present of 8-stage continuous
Runge-Kutta methods of uniform order 5 are the rst of their kind.
Title: Derivation of Ecient Continuous Explicit Runge-Kutta methods
Authors: B Owren and M Zennaro
Reference: SIAM J. Sci. Stat. Comp., 13 (1992), 1488{1501.
Reference: Tech. Rep. 240/90 (1990), Dept. Comp. Sci., Univ. Toronto.
Abstractz: Continuous Explicit Runge-Kutta methods with the minimal number of stages are considered. These
methods are continuously di erentiable if and only if one of the stages is the FSAL evaluation. A complete
characterization of these methods is developed for order 3, 4 and 5. It is shown how the free parameters of these
methods can be used either to minimize the continuous truncation error coecients or to maximize the stability
region. As a representative for these methods the 5th order method with minimized error coecients is chosen,
supplied with an error estimation method, and analysed by using the DETEST software. The results are compared
to a similar implementation of the Dormand-Prince 5(4) pair with interpolant. A possible interpretation of these
results is that the new method is about 40% more ecient than the continuous Dormand-Prince method.
Title: A Smoother Interpolant for DE/STEP, INTRP and DEABM
Authors: L F Shampine and H A Watts
Reference: Tech. Rep. SAND83{1226 (1983), Sandia Nat. Labs., USA.
Abstract: This report discusses a way of obtaining a smoother (globally C1) interpolant for the Adams family of
ODE methods. Details are provided for implementing the algorithm in the DE/STEP, INTRP suite of codes and
its descendant DEABM.
5

Title: Interpolation for Runge-Kutta Methods
Authors: L F Shampine
Abstract: Runge-Kutta methods provide a popular way to solve the initial value problem for a system of ordinary
di erential equations. In contrast to the Adams methods, there is no natural way to approximate the solution
between meshpoints. A way to accomplish this is proposed which is applicable to some important formulae. Its
theoretical support is much better than that of interpolation in the popular variable order, variable step Adams
codes.
Title: Interpolation for Variable Order Runge-Kutta Methods
Authors: L F Shampine
Reference: Comput. Math. Applic., 14 (1987), 255{260.
Abstract: Runge-Kutta methods approximate the solution of an initial value problem for a system of ordinary
di erential equations only on a mesh. Recent work considers how to approximate the solution everywhere. Variable
order, Runge-Kutta methods are developed that have this capability. One is about as ecient as a popular variable
order method without this capability.
Title: Interpolation and Error Estimation in Adams PC-Codes
Authors: H J Stetter
Abstract: Adams PC-methods form the basis of some of the most powerful codes for initial value problems in
ODEs. Several problems arising is such codes in connection with interpolation and local error estimation are
discussed analytically and algorithmically. Solutions are suggested and the experiences from extensive experiments
are reported. The results are relevant for global error estimation procedures for such codes.
Title: Multistep High Order Interpolants of Runge-Kutta Methods
Authors: R Vermiglio
Reference: Tech. Rep. UDMI/28/90/RR (1990), Dept. Math. Comp. Sci., Univ. Udine.
Abstractz: We consider a p-th order Runge-Kutta method
K(n)
i = f(xn + cih;yn +h
Pv
j=1aijK(n)
j ) i = 1;...;v
yn+1 = yn +h
Pv
i=1 biK(n)
i
for solving an initial value problem for ordinary di erential equations. The aim of this paper is to construct p-th
order interpolants by using the values furnished by the method on N successive intervals of integration. By using
Lagrange interpolation one can obtain a p-th order interpolant over p intervals, but we are interested in nding the
minimumnumber of intervals needed to obtain this.
We provide the conditions to satisfy and, in particular, we consider the class of collocation methods to obtain an
estimation of the number N. Some examples are given.
Title: Multistep Natural Continuous Extensions of Runge-Kutta Methods: the Potential for Stable Interpolation
Authors: R Vermiglio and M Zennaro
Reference: Appl. Numer. Math., 12 (1993), 521{546.
Abstract: The present paper develops a theory of multistep natural continuous extensions of Runge-Kutta meth-
ods, that is interpolants of multistep type that generalize the notion of natural continuous extensions introduced
by Zennaro. The main motivation for the de nition of such a type of interpolant is given by the need for interpo-
lation procedures with strong stability properties and a high-order of accuracy, in view of interesting applications
6

to the numerical solution of delay di erential equations and to the waveform relaxation methods for large systems
of ordinary di erential equations.
Title: A Smoother Interpolant for DE/STEP, INTRP and DEABM: II
Authors: H A Watts
Reference: Tech. Rep. SAND84{0293 (1984), Sandia Nat. Labs., USA.
Abstract: This report presents an improved smoother interpolant for the Adams family of ODE methods. A
correction to the STEP code is given which eliminatesan error that occurred under certain circumstances in forming
the higher order coecients for the method. Listings and details are provided for implementing the interpolation
algorithm in the DE/STEP, INTRP suite of codes, and its descendant DEABM.
Title: Smoother Interpolants for Adams Codes
Authors: H A Watts and L F Shampine
Reference: SIAM J. Sci. Stat. Comput., 7 (1986), 334{345.
Abstract: The Adams family of ODE methods is based on polynomial interpolants to past values obtained on a
discrete mesh. It is desirable, and in some circumstances essential, to have an interpolation scheme which produces
globally continuous approximations to both the solution and its derivative. In this paper we describe several ways
of achieving the desired result in an ecient manner, and we give particular emphasis to the task of achieving
mathematical continuity by the computational algorithm.
Title: Some Relationships between Implicit Runge-Kutta, Collocation and Lanczos Methods, and their Stability
Authors: K Wright
Reference: BIT, 10 (1970), 217{227.
Abstract: In this paper relationships between various one-step methods for the initial value problem in ordinary
di erential equations are discussed and a uni ed treatment of the stability properties of the methods are given. The
analysis provides some new results on stability as well as alternative derivations for some known results. The term
stability is used in the sense of A-stability as introduced by Dahlquist. Conditions for any polynomial collocation
method or its equivalent to be A-stable are derived. These conditions may be easily checked in any particular case.
Title: One-step Collocation: Uniform Convergence, Predictor-Corrector method, Local Error Estimate
Authors: M Zennaro
Abstract: In this paper we consider the one-step collocation method at Gaussian points for the numerical solution
of IVP's for ODE's. It is well-known that this method has a superconvergence rate O(h2n) at the nodes, while
it uniformly converges with the lower order O(hn+1). We give a rule to reach the uniform superconvergence by
some kind of IDeC-method. An interesting by-product of this procedure is an accurate estimation of the local
discretization error. Moreover we give a modi ed version of the method in a predictor-corrector mode, for which
all the previous results keep on holding; in particular we propose a new method of order four with stepsize control.
Title: Natural Continuous Extensions of Runge-Kutta Methods
Authors: M Zennaro
Reference: Tech. Rep. 80 (1984), Inst. Math., Univ. Trieste.
Abstract: The present paper develops a theory of Natural Continuous Extensions (NCEs) for the discrete ap-
proximate solution of an ODE given by a Runge-Kutta process. These NCEs are de ned in such a way that the
continuous solutions furnished by the one-step collocation methods are included.
7

Title: Natural Runge-Kutta and Projection Methods
Authors: M Zennaro
Abstract: Recently the author de ned the class of natural Runge-Kutta methods and observed that it includes all
the collocation methods. The present paper is devoted to a complete characterization of this class and it is shown
that it coincides with the class of the projection methods in some polynomial spaces.
2 Delay Di erential Equations
2.1 Numerical Methods
Title: Fixed Step Discretization Methods for Delay Di erential Equations
Authors: K Allen and S McKee
Reference: Comp. Math. Applic., 7 (1981), 413{423.
Abstract: The convergence of a fairly wide class of xed step linear discretization methods for delay di erential
equations with variable delay is studied. A global order convergence result which permits the case when disconti-
nuities are present is given. This utilizes a new formulation of the root condition for Dahlquist stability. Numerical
results illustrating the theory are presented.
Title: The In uence of Interpolation on the Global Error in Retarded Di erential Equations
Authors: H Arndt
Reference: Di erential-Di erence Equations, ISNM 62, 9{17, Birkhauser 1983.
Reference: Tech. Rep. 545 (1982), Sond. Approx. Optim., Univ. Bonn.
Abstract: Retarded initialvalue problems are routinely replaced by an initialvalue problemof ordinary di erential
equations along with an appropriate interpolation scheme. Hence one can control the global error of the modi ed
problem but not directly the actual global error of the original problem. In this paper we give an estimate for the
actual global error in terms of controllable quantities. Further we show that the notion of local error as inherited
from the theory of ordinary di erential equations must be generalized for retarded problems. Along with the new
de nition we are led to developing a reliable basis for a step selection scheme.
Title: Numerical Solution of Retarded Initial Value Problems: Local and Global Error and Stepsize Control
Authors: H Arndt
Abstract: Retarded initialvalue problems are routinely replaced by an initialvalue problemof ordinary di erential
equations along with an appropriate interpolation scheme. Hence one can control the global error of the modi ed
problem but not directly the actual global error of the original problem. In this paper we give an estimate for the
actual global error in terms of controllable quantities. Further we show that the notion of local error as inherited
from the theory of ordinary di erential equations must be generalized for retarded problems. Along with the new
de nition we are led to developing a reliable basis for a step selection scheme.
Title: Numerical Integration of Retarded Di erential Equations with Periodic Solutions
Authors: H Arndt, P J van der Houwen and B P Sommeijer
Reference: Delay Equations, Approximation and Application, ISNM 74, 41{51, Birkhauser 1985.
Reference: Tech. Rep. NM-R8502 (1985), Centre Math. Comp. Sci., CWI.
Abstract: It is the purpose of this paper to show that the minimaxversions of linear multistep methods, originally
derived for ordinary di erential equations with a periodic solution, are also suitable for the integration of retarded
8

di erential equations possessing a periodic solution. Especially for this type of equation, it is extremely useful to
have methods yielding highly accurate results for relatively large time steps h. We consider several examples of
rst-order and second-order equations with constant and state-dependent delay and compare the numerical results
with that of the conventional methods.
Title: Experience of STRIDE applied to delay di erential equations
Authors: C T H Baker, J C Butcher and C A H Paul
Abstract: STRIDE is intended as a robust adaptive code for solving initial value problems for ordinary di erential
equations (ODEs). The acronym STRIDE stands for STable Runge-Kutta Integrator for Di erential Equations.
Our purpose here is toreport onits adaptation forthe numericalsolutionofa test set of delayand neutral di erential
equations.
Title: Parallel continuous Runge-Kutta methods and vanishing lag delay di erential equations
Authors: C T H Baker and C A H Paul
Reference: Adv. Comp. Math., 1 (1993), 367{394.
Abstract: We present an explicit Runge-Kutta scheme devised for the numerical solution of delay di erential
equations (DDEs) where a delayed argument lies in the current Runge-Kutta interval. This can occur when the
lag is small relative to the stepsize, and the more obvious extensions of the explicit Runge-Kutta method produce
implicit equations. It transpires that the scheme is suitable for parallel implementation for solving both ODEs and
more general DDEs. We associate our method with a Runge-Kutta tableau, from which the order of the method can
be determined. Stability will a ect the usefulness of the scheme and we derive the stability equations of the scheme
when applied to the constant-coecient test DDE u0(t) = u(t)+u(t?), where the lag and the Runge-Kutta
stepsize Hn H are both constant. (The case = 0 is treated separately.) In the case that 6= 0, we consider the
two distinct possibilities: (i) H and (ii) H.
Title: A Global Convergence Theorem for a Class of Parallel Continuous Explicit Runge-Kutta Methods and
Vanishing Lag Delay Di erential Equations
Reference: SIAM J. Numer. Anal., awaiting publication.
Abstract: Iterated continuous extensions (ICEs) are continuous explicit Runge-Kutta methods developed for the
numerical solution of evolutionary problems in ordinary and delay di erential equations (DDEs). ICEs have a
particular role in the explicit solution of DDEs with vanishing lags. They may be regarded as parallel continuous
explicit Runge-Kutta (PCERK) methods, as they allow advantage to be taken of parallel architectures. ICEs can
also be related to a collocation method.
The purpose of this paper is to provide a theorem giving the global order of convergence for variable-step imple-
mentations of ICEs applied to state-dependent DDEs with and without vanishing lags. Implications of the theory
for the implementation of this class of methods are discussed and demonstrated. The results establish that out
approach allows the construction of PCERK methods whose order of convergence is restricted only by the continuity
of the solution.
Title: Issues in the numerical solution of evolutionary delay di erential equations
Authors: C T H Baker, C A H Paul and D R Will
e
Reference: Adv. Comp. Math., 3 (1995), 171{196.
9

Abstract: Delay di erential equations are of sucient importance in modelling real-life phenomena to merit the
attention of numerical analysts. In this paper, we discuss key features of delay di erential equations (DDEs) and
consider the main issues to be addressed when constructing robust numerical codes for their solution. We provide
an introduction to the literature and numerical codes, and in particular we indicate the approaches adopted by the
authors. We also indicate some of the unresolved issues in the numerical solution of DDEs.
Title: Parameter Estimation and Identi cation for Systems with Delays
Authors: H T Banks, J A Burns and E M Cli
Reference: SIAM J. Control Optim., 19 (1981), 791{828.
Abstract: Parameter identi cation problems for delay systems motivated by examples from aerodynamics and
biochemistry are considered. The problem of estimation of the delays is included. Using approximationresults from
semigroup theory, a class of theoretical approximationschemes is developed and two speci c cases (averaging and
spline methods) are shown to be included in this treatment. Convergence results, error estimates, and a sample
of numerical ndings are given.
Title: Estimation of Delays and other Parameters in Nonlinear Functional Di erential Equations
Authors: H T Banks and P K D Lamm
Reference: SIAM J. Control Optim., 21 (1983), 895{915.
Abstract: We discuss a spline-based approximation scheme for nonlinear non-autonomous delay di erential equa-
tions. Convergence results (using dissipative type estimates on the underlying nonlinear operators) are given in the
context of parameter estimation problems which include estimation of multiple delays and initial data as well as
the usual coecient-type parameters. A brief summary of some of our related numerical ndings is also given.
Title: On the Equivalent Bilinearization of Non-Linear Controlled Delay Systems
Authors: S P Banks
Reference: Int. J. Systems Sci., 17 (1986), 1389{1398.
Abstract: Taylor polynomials are used to obtain an equivalent bilinear system for a non-linear analytic delay
equation with control and the optimal control of the resulting bilinear system is considered.
Title: One-Step Collocation for Delay Di erential Equations
Authors: A Bellen
Abstract: Consider the following initial value problem for the delay di erential equation
y0(t) = f(t;y(t);y(t ? (t))); t0 t x; 0 (t) r;
y(t0) = y0; y(t) = (t); t0 ? r t t0;
where y, and f are m-vector valued functions and is a piecewise continuous scalar function. ...... This paper
deals with the one-step collocation method with continuous piecewise polynomial functions of degree n 1 and
collocation points at the zeroes of the n-Legendre orthogonal polynomial relative to each subinterval between two
consecutive nodes of the mesh = (t0;...;tN = x). Results on convergence and superconvergence are proved
for appropriate choices of the mesh and f smooth enough. Namely the global error is O(jjn+1) and the error
at the nodes of the mesh is O(jj2n) provided the mesh and f are such that the solution belongs to C2n+1
(i.e.
f 2 C2n[ti?1;ti], i = 1;...;N).
Title: A Runge-Kutta-Nystrom Method for Delay Di erential Equations
Authors: A Bellen
Reference: Progress Sci. Comp., 5 (1985), 271{283.
10

Abstractz: Let us consider the following boundary value problem for second-order delay di erential systems:
y00(t) = f(t;y(t);y0(t);y(t ?(t));y0(t?(t))) t0 t b
y(t) = (t) t t0;
y0(t) = 0(t) t t0;
y(b) = yb
y : R! Rm, f : [t0;b] R4m ! R and (t);(t) 0. ...... Here we treat the problem by means of collocation in
piecewise polynomial spaces either in the shooting or in the global approach. In both cases, under some conditions
on the delays and on the choice of the mesh, the superconvergence phenomenon, well-known for ODEs, extends to
DDEs.
Alternatively, in the shooting approach, an iterated-defect-correction like methods is presented which allows to treat
some more general cases and to drop the conditions on the choice of mesh.
Title: Numerical Solution of Delay Di erential Equations by Uniform Corrections to an Implicit Runge-Kutta
Method
Abstract: In this paper we develop a class of numerical methods to approximate the solutions of delay di erential
equations. They are essentiallybased on amodi edversion, inapredictor-corrector mode,ofthe one-step collocation
method at n Gaussian points. These methods, applied to ODEs, provide a continuous approximatesolution which is
accurate to order 2n at the nodes and of order n+1 uniformlyin the whole interval. In order to extend the methods
to delay di erential equations, the uniform accuracy is raised to the order 2n by some a posteriori corrections.
Numerical tests and comparisons with other methods are made on real-life problems.
Title: A Survey of the Mathematical Theory of Time-Lag, Retarded Control, and Hereditary Processes
Authors: R Bellman and J M Danskin
Reference: Tech. Rep. R-256 (1954), Rand Corporation, Santa Monica.
Abstract: This report contains a summary of the mathematical techniques required to treat physical phenomena
involving time lags, retarded control, or hereditary e ects. The functional equations which arise are no longer the
di erential equations of classical mathematical physics, but rather di erential-di erence equations, integrodi eren-
tial equations, and equations of even more complicated form.
The most important applications of the mathematical theory arise in connection with control problems, and the
resulting stability investigations. Questions involving these advanced methods arise in the theory of guided missiles
and pilotless aircraft. It is here that the tremendous velocities, which are now feasible, make the time lags created
by the control mechanism of great signi cance. In many cases it is impossible even to understand the origin of
various instability phenomena without taking into account time lags and retarded control. For this reason these
ideas are becoming of increasing importance in the eld of servomechanisms and automatic control.
Equations of this form play an important role in mathematical economics where the analysis of interindustrial
processes requires an awareness of the fact that some changes cannot occur instantaneously.
Other physical phenomena requiring these concepts occur in the theory of magnetism,in the theory of elasticity and
plasticity, and throughout the theory of ssion processes. In the led of biology these ideas are required to explain
mutation and, in general, the growth of unicellular organisms. They are of particular importance in furnishing an
understanding of the e ects of radioactive exposure, and thus, indirectly, in the study of cancer. In the eld of
psychology, they are necessary for any treatment of learning theory and other long-term e ects, such as mental
breakdown.
11

Problems of thus type will occur where the future depends not only on the immedaite present, but also on the past
history of the system under consideration.
Title: Numerical Integration of a Di erential-Di erence Equation with a Decreasing Time-Lag
Authors: R E Bellman, J D Buell and R E Kalaba
Reference: Comm. ACM, 8 (1965), 227{228.
Abstract: Systems in which variable time-lags are present are of common occurrence in biology. Variable ow
rates are a common cause of these lags. At present no extensive body of knowledge exists concerning the e ects
which these variable lags can cause. Shown here is a method of reducing some di erential-di erence equations to
ordinary di erential equations which can then be studied numerically with ease. Subsequent study will deal with
situations in which multiple-lags and lags dependent on the solution itself are present.
Title: Mathematical Experimentation in Time-Lag Modulation
Authors: R E Bellman, J D Buell and R E Kalaba
Reference: Comm. ACM, 9 (1966), 752{754.
Abstract: Equations of the form du=dt = g(u(t);u(h(t))) arise in a number of scienti c contexts. The authors
point out some interesting properties of the solution of u0(t) = ?u(t ? 1 ? ksin(!t)) + sin( t). These properties
were obtained by means of numerical solution.
Title: On Global Methods to Solve Di erence-Di erential Equations with Retarded Arguments
Authors: A Bleyer
Reference: Periodica Poly., 22 (1978), 141{148.
Abstract: In the recent time interest is focussed on investigations into di erence-di erential equations, because
of the wide scale of applications. Nevertheless the numerical methods for solving this kind of equations have not
developed so far. Here it is attempted to nd a general theorem for the equation y0(t) = F(t;y(t);y(t?!)) (! 0),
and that the convergence of a spline method. Finallysome problemsare discussed concerning the numericalstability
of the presented method.
Title: Numerical Computation of Optimal Controls in the Presence of State-Dependent Time Lags
Authors: H G Bock and J P Schloder
Reference: Proc. Int. Fed. Autom. Control, (1984), Pergamon.
Abstract: The present paper treats several numerical aspects of an algorithm designed for a very general class
of optimal control problems with state-dependent time delays in control and state variables. The approach chosen
deals with computation of parameterized controls by successive quadratic approximation. The emphasis of the
paper is put on the key problem of solving the dynamical systems with delays and associated variational systems in
an ecient, safe and stable way by a variable order and step multistep method REBUS, based on the concept of
natural interpolation. As a numerical application, the optimal control of a chemical reactor from a perturbed to an
equilibrium state is given.
Title: The Adaptation of STRIDE to Delay Di erential Equations
Authors: J C Butcher
Abstractz: The di erential equation solver STRIDE is at present being modi ed to make it more convenient to
use, especially for dicult and non-standard problems. In particular, its possible use for delay di erential equations
is being taken into account in the design.
What seems to be needed is a means of calculating approximate solution values outside a current step by interpola-
tion frominformationgenerated when previous steps were being worked through. In a solver of conventional design,
12

there is also needed a means of making this data from previous steps available to the subroutine which computes
the right hand side of the delay di erential equation.
In STRIDE, interpolation within a current step is carried out using an expansion in a modi ed form of Taylor
series, in which Laguerre polynomials take the place of the usual powers of the o set from the beginning of the
current step. This is inappropriate for later use in computing delays and part of the content of the paper deals
with a transformation of the data to a more convenient form. The other main topic discussed will be the use
of reverse communication to allow exibility of usage and to allow the extension to delay problems particularly
straightforward.
Title: The Application of Shifted Legendre Polynomials to Time-Delay Systems and Parameter Identi cation
Authors: R Y Chang and M L Wang
Reference: J. Dynamic Syst. Measur. Control { Trans. ASME 107 (1985), 79{85.
Abstract: A linear time-delay state equation is solved by the proposed shifted Legendre polynomialsmethod. The
parameter identi cation of such a system with time-delay is also studied. The system is partitioned into several
time intervals. Within a certain time interval, the state and control functions are assumed to be expressed by
the shifted Legendre polynomial series. Time-delay di erential equations are transformed into a series of algebraic
equations of expansion coecients. An e ective algorithm is proposed to solve the time-delay system problem and
to estimate the system parameters. Only a small number of leading terms of expansion coecients is enough to
get accurate results. By using such an e ective computational algorithm, the calculation procedures are greatly
simpli ed. Thus much computer time is saved.
Title: Time Lag Systems { A Bibliography
Authors: N H Choksy
Reference: IRE Trans. Autom. Control, AC-5 (1966), 66{70.
Abstract: In a recent issue of Transactions on Automatic Control, Weiss has given an excellent annotated bibliog-
raphy on the subject of transportation lag. He was kind enough to refer to a previous bibliography which appeared
in this author's thesis. Since the latter is not generally available, the bibliography therein is given here. The items
which have already appeared in Weiss' listing have been omitted; items which have appeared since the thesis was
written have been included.
Whilst most of the items here were found by library searches, acknowledgement must be made to Bellman's bibli-
ographies as a source.
If there are items which have not been mentioned either here or in Weiss' paper, the author of the present paper
would appreciate being informed about it by either a complete reference or a copy of the paper (or papers) in
question.
A short introduction on the mathematical characterization of time lag systems is given before the bibliography.
Title: Highly-stable Multistep methods for Retarded Di erential Equations
Authors: C W Cryer
Abstract: A linear multistep method (;) is de ned to be DA0-stable if when it is applied to the delay di erential
equation y0(t) = ?y(t ? ) the approximate solution yh(tn) ! 0 as n ! 1 for all 2 (0;=2) and all stepsizes h
of the form h = =m, m a positive integer.
General properties of DA0-stable methods are derived. These properties are similar to the properties of A-stable
and A( )-stable methods; for example, it is proved that a k-step DA0-stable method of order k must be implicit.
As an application it is shown that the trapezoidal method is DA0-stable.
Finally, the condition that h = =m is dropped and the resulting methods, which we call GDA0-stable methods,
are studied.
13

Title: Numerical Solution of Ordinary and Retarded Di erential Equations with Discontinuous Derivatives
Authors: A Feldstein and R Goodman
Abstract: This paper studies the propagation of discretization error for discontinuous ordinary and retarded
di erential equations. Various applications are given including one which extends a fundamentaltheorem of Henrici
concerning round-o error.
Title: High Order Methods for State-Dependent Delay Di erential Equations with Non-Smooth Solutions
Authors: A Feldstein and K W Neves
Abstract: This work presents a theoretical basis for high-order numerical methods to solve state-dependent delay
di erential equations of the form
x0(t) = f(t;x(t);x( (t;x(t)))) for t 2 [a;b];
(t;x(t)) t;
x(t) = (t) for t 2 [
a;a];
where
a = min (t;x(t)) for t 2 [a;b]. The solutions to such equations typically have derivative jump discontinuities
(jump points) which propagate from the initial jump point t = a. Thus, high-order methods require an accurate
determination of the location of jump discontinuities in lower order derivatives of the solution x(t). The locations
of these jump points are characterized as the zeros of certain nonlinear equations which themselves depend on
x(t). Because of this interdependence, the following non-trivial question arises: Can these unknown jump points be
determined accurately enough to develop high-order methods? This question is resolved armatively by exploring
special properties of delay di erential equations.
Title: Numerical Studies and Sharpness Results for State-Dependent Delay Di erential Equations
Authors: A Feldstein and K W Neves
Abstract: The purpose of this paper is to present and discuss several numericalstudies that illustrate the sharpness
of order theorems in previous papers by the authors, and demonstrate certain computational advantages of the
authors' restart method over the derivative correction approach of Hutchinson and Zverkina. These studies also
point out various subtle nuances of the theory that have practical importance.
Title: Stepsize Control for Delay Di erential Equations using a Pair of Formulae
Authors: S Filippi and U Buchacker
Abstract: The e ect of the local approximation error on the stepsize control at one-step methods, which are used
for the numerical solution of delay di erential equations are considered. It is shown, how to get a reliable estimate
for the local error and so a working stepsize control by using a pair of formulae.
Title: Round-o Error for Retarded Ordinary Di erential Equations: A Priori Bounds and Estimates
Authors: R Goodman and A Feldstein
Abstract: Consider the numerical solution of a retarded ordinary di erential equation (RODE) by some standard
algorithms. For a linear RODE, we estimate the accumulated round-o error as a linear combination of the
preceding local round-o errors, and we bound the accumulated round-o error. For a non-linear RODE, we obtain
by linearization similar estimates and bounds for the dominant part of the accumulated round-o error.
Title: Error Control for Initial Value Problems with Discontinuities and Delays
Authors: D J Higham
14

Reference: Tech. Rep. NA/129 (1991), Dept. Math. Comp. Sci., Univ. Dundee.
Abstractz: When using software for ordinary di erential equation (ODE) initialvalue problems,it is not unreason-
able to expect the global error to decrease linearly with the user-supplied tolerance. For standard ODEs, conditions
on an algorithmthat guarantee such `tolerance proportionality'asymptotically(as the error tolerance tends to zero)
were derived by Stetter. Here we extend the analysis to cover a certain class of ODEs with low order derivative
discontinuities, and the class of ODEs with constant delays. We show that standard error control techniques will be
successful if discontinuities are crossed with suciently small steps and delay terms are calculated with suciently
accurate interpolants. It is perhaps surprising that several delay ODE algorithms that have been proposed do not
use suciently accurate interpolants to guarantee asymptotic proportionality. Our theoretical results are illustrated
numerically.
Title: A New Interpolation Procedure for Adapting Runge-Kutta Methods to Delay Di erential Equations
Authors: K J in't Hout
Reference: BIT, 32 (1992), 634{649.
Reference: Tech. Rep. TW-90-09 (1991), Dept. Math. Comp. Sci., Leiden Univ.
Abstract: This paper deals with adapting Runge-Kutta methods to di erential equations with a laggingargument.
A new interpolationprocedure is introduced which leads to numericalprocesses that satisfyan importantasymptotic
stability condition related to the class of test problems U0(t) = U(t) + U(t ? ) with ; 2 C, () ?jj,
and 0. If ci denotes the i-th abscissa of a given Runge-Kutta method, then in the n-th step tn?1 ! tn :=
tn?1 + h of the numerical process our interpolation procedure computes an approximation to U(tn?1 + cih ? )
from approximations that have already been generated by the process at points tj?1+cih (j = 1;2;3;...). For two
of these new processes and a standard process we shall consider the convergence behaviour in an actual application
to a given, sti problem.
Title: Analysis and Parameter Identi cation of Time-Delay Systems via Shifted Legendre Polynomials
Authors: C Hwang and M Y Chen
Reference: Int. J. Control, 41 (1985), 403{415.
Abstract: The problems of analysis and parameter identi cation of time-delay systems have been recently studied
using rectangular functions such as Walsh and block-pulse functions. In this paper, the continuous shifted Legendre
polynomials are rst used to solve the above mentioned problems. The approach followed is that of converting
the delay di erential equation to an algebraic form through using the operational matrices of integration and
delay. The key point is the derivation of a delay operational matrix D which relates the coecient vector x of the
shifted Legendre series of a given function x(t) with the coecient vector x of its delay form x(t?) in the form
x = Dx+ , where the elements of are obtained from the initial function x(t). Two computational examples
are given to illustrate the utility of the method. The availability of recursive algorithms for constructing the delay
operational matrix D makes the method particularly attractive to practice.
Title: One-Step Methods for Retarded Di erential Equations with Parameters
Authors: T Jankowski
Reference: Computing, 43 (1990), 343{359.
Abstract: In this paper we develop a class of one-step methods combined with iterative methods to approximate
the solution of retarded di erential equations with parameters. The convergence theorems are established including
the estimation of errors. Some methods are illustrated by three numerical examples.
Title: Computer Solutions of State-Dependent Delay Di erential Equations
Authors: A Karoui and R Vaillancourt
15

Reference: Comput. Math. Applic., 27 (1994), 37{51.
Abstract: An adaptation of the Runge-Kutta-Verner (5,6) formulapair is used to construct a numericalmethod for
the solution of state-dependent delay di erential equations with nonvanishing lag. A fth-degree divided-di erence
Newton backward interpolation polynomial is used to nd the location of derivative jump discontinuities of the
solution. In order to maintain the sixth-order accuracy of the RKV pair, the value of the solution at the delay
is approximated by a three-point Hermite polynomial. This new method is tested on some real-life problems. A
Fortran program, called SYSDEL, is available from the authors.
Title: Approximation Theory and Numerical Methods for Delay Di erential Equations
Authors: G Meinardus and G N
urnberger
Reference: Delay Equations, Approximation and Application, ISNM 74, 13{40, Birkhauser 1985.
Reference: Tech. Rep. 63 (1985), Fakul. Math. Infor., Mannheim.
Abstract: The latest research shows that the numerical treatment of delay di erential equations leads to various
approximationand optimizationproblems. In this manuscript of introductory character several aspects of this circle
of problems are described. Moreover, some problems are stated and a few new observations are added.
Title: Estimationof Time- and State-Dependent Delays and other Parameters in Functional Di erential Equations
Authors: K A Murphy
Reference: SIAM J. Appl. Math., 50 (1990), 972{1000.
Abstract: A parameter estimation algorithm is developed that can be used to estimate unknown time- or state-
dependent delays and other parameters (e.g., initial conditions) appearing within a nonlinear non-autonomous
functional di erential equation. The original in nite-dimensional di erential equation is approximated using linear
splines, which are allowed to movewith the variable delay. The variable delays are approximated using linear splines
as well. The approximation scheme produces a system of ordinary di erential equations with nice computational
properties. The unknown parameters are estimated within the approximatingsystems by minimizinga least-squares
t-to-data criterion. Convergence theorems are proved for time-dependent delays and state-dependent delays within
two classes, which say essentially that tting the data by using approximations will, in the limit, provide a t to
the data using the original system. Numerical test examples are presented that illustrate the method for all types
of delay.
Title: Automatic Integration of Functional Di erential Equations: An Approach
Authors: K W Neves
Reference: ACM Trans. Math. Soft., 1 (1975), 357{368.
Abstract: Heretofore, the automatic numerical integration of functional di erential equations (FDEs) has received
little attention. However, in recent years the use of such equations in the mathematical modelling of various
biological, environmental, and societal processes has increased markedly. The computer solution of FDEs of the
following form are considered:
X0(t) = f(t;X(t);Z(t)) (1)
for a t b (f a vector-valued function) where X(t) = (X1(t);...;Xn(t)), Z(t) = (X1( 1(t;X(t)));...;Xn( n(t;X(t))))
and i(t;X(t)) t for i n, and where the required initial functions are Xi(t) = i(t) for t a.
An approach to the conversion of automatic ordinary di erential equation (ODE) solvers to automatic FDE solvers
is considered. The approach emphasizes the preservation of the essential characteristics of the original ODE solver,
such as error estimation and step changing. The use of such a converted algorithm, which is capable of integrating
even more general FDEs than (1), is discussed. The ability of the algorithm to overcome the problem of derivative
discontinuities in the solution is also, discussed. Test problems presented include systems of FDEs with multiple
and vanishing lags, Volterra integro-di erential equations, and FDEs with nonsmooth solutions.
16

Title: Numerical Treatment of Delay Di erential Equations by Hermite Interpolation
Authors: H J Oberle and H J Pesch
Abstract: A class of numerical methods for the treatment of delay di erential equations is developed. These
methods are based on the well-known Runge-Kutta-Fehlberg methods. The retarded argument is approximated by
an appropriate multipoint Hermite interpolation. The inherent jump discontinuities in the various derivatives of
the solution are considered automatically.
Problems with piecewise continuous right-hand side and initialfunction are treated too. Real-life problems are used
for the numerical test and a comparison with other methods published in the literature.
Title: DELAY1 { A Program for Integrating Systems of Delay Di erential Equations
Authors: J Oppelstrup
Reference: Tech. Rep. TRITA-NA-7311 (1973), Dept. Info. Process. Comp. Sci., Roy. Inst. Tech. Stockholm.
Abstract: The problem is to obtain a numerical solution to functional di erential equations of retarded time delay
type. The method used is a straightforward application of the continuation procedure. This is most easily described
by means of a simple example.
Title: The RKFHB4 Method for Delay Di erential Equations
Authors: J Oppelstrup
Reference: Springer Lect. Note Math., 631 (1976), 133{146.
Abstract: A method for the numerical solution of initial value problems for systems of retarded delay di erential
equations is described. It uses a variable step Runge-Kutta Fehlberg 4/5 method combined with fourth degree
Hermite-Birkho interpolation, which makes the usual local error estimator asymptotically correct. To obtain good
numerical stability, the basic interpolation is modi ed for small delays. The possible initial discontinuities are
algorithmically treated by using a very short stepsize in the critical step, and to this end the stepsize control has
been somewhat modi ed. Numerical examples are included.
Title: Developing a Delay Di erential Equation Solver
Authors: C A H Paul
Abstract: We discuss brie y various phenomena to be tested when designing a robust code for delay di erential
equations: choice of interpolant; tracking of discontinuities; vanishing delays; and problems arising from oating
point arithmetic.
Title: Performance and Properties of a Class of Parallel Continuous Explicit Runge-Kutta Methods for Ordinary
and Delay Di erential Equations
Authors: C A H Paul
Abstract: In this paper we examine the parallel implementation of the iterated continuous extensions (ICEs) of
Paul Baker. We indicate how to construct arbitrarily high-order ICEs, and discuss some of the strategies for
choosingthe `free' parameters ofthe methods. The numericalresults that we present suggest that ICEs implemented
in parallel provide an easy and ecient way of writing dense-output ordinary di erential equation codes and delay
di erential equation codes, even in the case that the lag vanishes and/or is state-dependent.
Title: Stepsize Control for Delay Di erential Equations using Continuously Embedded Runge-Kutta Methods of
Sarafyan
17

Authors: S Thompson
Abstract: The use of continuously embedded Runge-Kutta-Sarafyan methods for the solution of ordinary di er-
ential equations with either time-dependent or state-dependent delays is discussed. It is shown how to get reliable
solutions for such problems in a manner that does not require that the e ect of the local approximationerror be con-
sidered separately from the local integration error. It is also shown how to reliably handle derivative discontinuities
that arise in the solution of di erential equations with delays.
Title: Explicit Solution of a Class of Delay Di erential Equations
Authors: A C Tsoi
Abstract: An explicit solution for a class of delay di erential equations is found in the form of an in nite series.
Methods have been given how to nd the coecients of this in nite series in the form of a recursive formula, or in
the form of a tree structure. Examples are given to illustrate the method.
Title: A One-step Subregion Method for Delay Di erential Equations
Authors: R Vermiglio
Reference: Calcolo, 22 (1986), 429{455.
Abstract: We study a one-step method for delay di erential equations, which is equivalent to an implicit Runge-
Kutta method. It approximates the solution in the whole interval with a piecewise polynomial of xed degree n.
For an appropriate choice of the meshpoints, it provides uniform convergence O(hn+1) and the superconvergence
O(h2n) at the nodes.
Title: The tracking of derivative discontinuities in systems of delay-di erential equations
Authors: D R Will
e and C T H Baker
Reference: Tech. Rep. 160/185 (1988/90), Dept. Math., Univ. Manchester.
Abstract: Evolutionary type delay-di erential equations occur widely in dynamical processes in many elds of
biology, engineering and the physical sciences. One of their characteristics is that their solutions frequently contain
derivative discontinuities. In this paper we propose a model for the tracking of such discontinuities and discuss
its in uenece on the design of numerical software. Our discussion is illustrated by DELSOL, a general purpose
delay-di erential equation solver developed at the University of Manchester.
Title: DELSOL { a numerical code for the solution of systems of delay-di erential equations
Authors: D R Will
e and C T H Baker
Reference: Appl. Numer. Math., 9 (1992), 223-234.
Abstract: Delay-di erential equations arise widely in many elds of science and engineering. In this paper we
present an overview of a general purpose code { DELSOL { written to solve systems of such equations. Special
attention is given to the program's organisation and design. A number of numerical examples are also presented
alongside a number of technical observations pertinent to the design of delay-di erential software.
Title: Stepsize control and continuity consistency for state-dependent delay-di erential equations
Authors: D R Will
e and C T H Baker
Abstract: Derivative discontinuities occur frequently in the solutions of delay-di erential equations, even when
18

the functions de ning them are all C1. Unless correctly treated, such discontinuities can undermine the continuity
assumptions made by ODE software. In this paper we discuss some speci c diculties arising from this aspect in
the treatment of state-dependent problems. A necessary property, continuity consistency, is introduced along with
a class of methods for which it is satis ed.
Title: Towards an Alternative Error Control Strategy for Ordinary and Delay Di erential Equations
Authors: D R Will
e
Abstract: Local error controls are widely used to control the global error in codes for the numerical solution
of ordinary and delay di erential equations. Sometimes, however, such methods may fail to take account of the
convergence and divergence properties of derivative elds. Here we propose an alternative technique based on a
rst-order linear variational analysis. Our ndings are presented for both ordinary and delay equations.
Title: Some Issues in the Detection and Location of Derivative Discontinuities in Delay Di erential Equations
Authors: D R Will
e and C T H Baker
Abstract: The detection and location of derivative discontinuities is a central issue in the design of robust solvers
for delay di erential equations. In this paper we discuss a number of particular features associated with one common
discontinuitytrackingstrategy before addressing aparticular problemarisingfor aclass ofstate-dependent problems.
2.2 Dynamics and (Numerical) Stability
Title: Fractal Basin Boundaries of a Delay Di erential Equation
Authors: J M Aguirregabiria and J R Etxebarria
Reference: Physics Letters A, 122 (1987), 241{244.
Abstract: It is pointed out that fractal boundaries between attraction basins of non-chaotic attractors appear also
in the case of a delay di erential equation. The nature and dimension of basin boundaries are analyzed in di erent
cases.
Title: Computing stability regions { Runge-Kutta methods for delay di erential equations
Reference: IMA J. Numer. Anal., 14 (1994), 347{362.
Abstract: We discuss the practical determination of stability regions when various xed-stepsize Runge-Kutta
(RK) methods, combined with continuous extensions, are applied to the linear delay di erential equation (DDE)
y0(t) = y(t) + y(t ? ) (t 0);
with xed delay . It is signi cant that the delay is not an integer multiple of the stepsize, and that we consider
various continuous extensions.
The stabilitylociobtained inpractice indicate that the standard boundary-locus technique (BLT) can failtomapthe
RK DDE stability region correctly. The aim of this paper is to present an alternative stability boundary algorithm
that overcomes the diculties encountered with the BLT. This new algorithm can be used for both explicit and
implicit RK methods.
Title: Dynamics of Discretized Equations for DDEs
Authors: C T H Baker
19

Abstract: In this note, we illustrate the observation that discretized delay equations can produce irregular dy-
namical behaviour similar to that seen in discretized evolutionary problems in ordinary di erential equations. Such
phenomena, associated with nonlinear problems, are a challenge to the robustness of the method.
Title: Delay Equations, the Left-Shift Operator and the In nite-Dimensional Root Locus
Authors: S P Banks and F Abbasi-Ghelmansarai
Abstract: The distributed parameter root locus is considered and a relation between simple delay equations and
the left-shift operator is developed. This gives a rigorous explanation of the s-plane behaviour of delay systems and
shows that the classical root locus starting on the open-loop poles of the systems can become bands swept out by
connected components of the spectrum of the system operator in the in nite-dimensional case.
Title: Existence of Periodic Solutions in n-Dimensional Retarded Functional Di erential Equations
Authors: S P Banks
Abstract: The existence of periodic orbits of n-dimensional delay systems of the form x0(t) = ?f(x(t ? p)) is
proved and applied to systems of the form x0(t) = ?x(t? 1)N(x(t)), and to a certain type of Hamiltonian system.
Title: Strong Contractivity Properties of Numerical Methods for Ordinary and Delay Di erential Equations
Reference: Appl. Numer. Mathe., 9 (1992), 321{346.
Reference: Tech. Rep. 224 (1990), Dipart. Sci. Mate., Univ. Trieste.
Abstractz: In the last twenty years various stability test problems for DDE solvers have been considered, which
are mainly generalizations of those used for ODEs. For the simplest autonomous case y0(t) = y(t) + y(t ? ),
the concepts of P- and GP-stability were introduced and a signi cant number of results have already been found
for both classes of linear multistep methods and one-step Runge-Kutta methods. More dicult is the situation
for the non-autonomous linear test equation y0(t) = (t)y(t) + (t)y(t ? ) and for the general nonlinear case
y0(t) = f(t;y(t);y(t ? )), which gave rise to the concepts of PN-, GPN-, RN- and GRN-stability. In particular,
in order to study PN- and GPN-stability, a new stability notion for ODE solvers based on the test equation with
forcing term y0(t) = (t)y(t) + f(t), which is called ANf-stability and BNf-stability, which are based on the test
equations y0(t) = y(t) + f(t) and y0(t) = f(t;y(t);u(t)) respectively. We give some relationships among all these
concepts of stability for ODEs and DDEs and outline the situation for the class of Runge-Kutta methods up to
order 2.
Title: P-Stable and P[ ; ]-Stable Integration/Interpolation Methods in the Solution of Retarded Di erential-
Di erence Equations
Authors: T A Bickart
Reference: BIT, 22 (1982), 464{476.
Abstract: The equation u0(t) = pu(t) +qu(t ?) is presented as an archetype (scalar) equation for assessing the
quality of integrator/interpolator pairs used to solve retarded di erential-di erence equations. The relationships of
P-stability and P[ ; ]-stability,de ned with respect to this archetype equation, to stability and order of multistep
integrators and to passivity and order of Lagrange interpolators are developed. Composite multistep integrators
and composite Lagrange interpolators are considered as a means of obtaining high-order pairs stable for all stepsizes
over a large portion, if not all, of the (p;q)-domain on which the archetype equation is stable.
Title: Perturbations and Delays in Di erential Equations
Authors: T A Burton
20

Reference: SIAM J. Appl. Math., 29 (1975), 422{438.
Abstract: In this paper we present a number of results on perturbations of second order di erential equations
of the form x00 + f(x)h(x0)x0 + g(x) = 0. This is accomplished by constructing a variety of Lyapunov functions.
We then show how these Lyapunov functions can be converted to Lyapunov functionals for the delay equation
x00 + f(x)h(x0)x0 + g(x(t ? (t))) = 0, thereby obtaining boundedness results. Some of the work is generalized to
higher order systems. We also present some continuation results for higher order delay equations. Several examples
are given.
Title: Oscillatory Solutions of x00(t)+ a(t)f(x(q(t))) = 0
Authors: T Burton and R Grimmer
Reference: Delay and Functional Di erential Equations and Their Applications, 335{344, Academic Press, New York
1972.
Abstract: We present here a number of results concerning the oscillatory behaviour of the solutions of the second
order retarded di erential equation
x00(t)+ a(t)f(x(q(t))) = 0: (2)
Also, we obtain a result which gives a necessary condition for a non-oscillatory solution x(t) of 2 to satisfy x(t)
m(t) 0 for large t, where m(t) in an arbitrarily chosen function. In particular, choosing m(t) to be a constant
function, we then obtain a sucient condition for all solutions to be oscillatory. Finally, we present some examples
to demonstrate the applicability of our results.
Title: The Oscillation and Exponential Decay Rate of Solutions of Di erential Delay Equations
Authors: Y Cao
Reference: Contemp. Math., 129 (1992), 43{54.
Abstract: This paper generalizes the discrete Lyapunov function to state-dependent delay di erential equations.
The discrete Lyapunov function is a measure of oscillation of solutions on intervals whose length equals the time
delay at the state being zero. It establishes a relation among the oscillation, the exponential decay rate and the
rst-order estimation of solutions which go to zero as t goes to positive in nity. It also proves that the faster a
solution oscillates, the faster it decays.
Title: A General Theory of Convergence for Numerical Methods
Authors: B A Chartres and R S Stepleman
Abstract: The concepts of convergence, consistency and stability are given very general de nitions that are
applicable to any numerical computation. The main feature of these de nitions is the treatment of truncation
error and rounding error as special cases of the unifying concept of a perturbation. Thus stability, which has to
do with rounding errors, and consistency, which has to do with truncation errors, are both expressed in terms of
perturbations. This gives an abstract version of the Lax theorem (consistency + stability = convergence) a very
simple proof, applicable to any numerical method. The conceptual identi cation of the two sources of error also
yields a general theory of the order of convergence of an algorithm in the presence of rounding errors. It is proved
that the order of convergence is the same as the order of consistency if the method has a property called order-
stability. The usefulness of the theory is demonstrated by analyses of methods for solving both initial value and
boundary value problems in ordinary di erential equations, where it is found that order-stability and stability are
equivalent. The theory does not obviate the need for any of the detailed analysis normally required, but the insight
it provides enables us to extend the traditional results to include di erential equations y0 = f(x;y(x)) in which f
is a discontinuous function of its rst argument.
21

Title: Existence and Stability of Solutions of a Delay Di erential System
Authors: R D Driver
Reference: Arch. Rational Mech. Anal., 10 (1962), 401{426.
Abstract: Di erential systems involving rather arbitrary retarded arguments have received much attention in the
Russian literature, particularly during the past 10 years or so. This is apparently due, in part, to the importance
of such systems in automatic control theory.
The present paper gives the basic existence theorems and an introduction to the Lyapunov method for a generalized
di erence-di erential system slightly more general than those usually considered in the Russian literature.
Title: An Asymptotic Analysis of the Delayed Logistic Equation when the Delay is Large
Authors: A C Fowler
Reference: IMA J. Appl. Math., 28 (1982), 41{49.
Abstract: We show how to construct an asymptotic solution to the delayed logistic equation y0(t) = y(1 ?
y1), corresponding to the asymptotic limit ! 1. The results of the analysis are compared with a numerical
computation, and found to be comparatively accurate for 2. Since the approach is novel, we comment on some
features which may be relevant to other problems.
Title: On a Functional Di erential Equation
Authors: L Fox, D F Mayers, J R Ockendon and A B Tayler
Reference: J. Inst. Math. Applic., 8 (1971), 271{307.
Abstract: This paper considers some analytical and numerical aspects of the problem de ned by an equation or
systems of equations of the type y0(t) = ay(t) + by(t), with a given initial condition y(0) = 1.
Series, integral representations and asymptotic expansions for y are obtained and discussed for various ranges of
the parameters a, b and 0, and for all positive values of the argument t. A perturbation solution is constructed
for j1 ? j 1, and con rmed by direct computation. For 1 the solution is not unique, and an analysis is
included of the eigensolutions for which y(0) = 0.
Two numerical methods are analysed and illustrated. The rst, using nite di erences, is applicable for 1, and
two techniques are demonstrated for accelerating the convergence of the nite-di erence solution towards the true
solution. The second, an adaptation of the Lanczos method, is applicable for any 0, though an error analysis
is available only for 1. Numerical evidence suggests that for 1 the method still gives good approximations
to some solution of the problem.
Title: Periodic Solutions of Lienard Equations with Delay: Some Theoretical and Numerical Results
Authors: R B Grafton
1972.
Abstract: We consider here the following Lienard equation with delay:
x00(t)+ f(x(t))x0(t) +g(x(t ?r)) = 0;r 0:
In the case r = 0, conditions on f(x) and g(x) for which the equation will have periodic solutions are well-known.
These conditions can be expressed in geometric terms , and proofs of the results can be geometrically motivated;
see for example [3]. If a delay is present, i.e. r 0, then proofs become more complicated. However, much of the
geometric motivation carries over to the delay case.
Title: Stability of Adaptive Algorithms for Delay Di erential Equations
Authors: D J Higham and I T Famelis
Reference: Tech. Rep. 146 (1992), Dept. Math. Comp. Sci., Dundee Univ.
22

Abstract: This work examines the performance of explicit, adaptive, Runge-Kutta based algorithms for solving
delay di erential equations. The results of Hall for ordinary di erential equation (ODE) solvers are extended by
adding a constant-delay term to the test equation. It is shown that by regarding an algorithmas a discrete nonlinear
map, xed points, or equilibrium states, can be identi ed, and their stabilitycan be determined numerically. Speci c
results are derived for a loworder Runge-Kutta pair coupled with either alinear or cubic interpolant. The qualitative
performance is shown to depend upon the interpolation process, in addition to the ODE formula and the error
control mechanism. Further, and in contrast to the case for standard ODEs, it is found that the parameters in the
test equation also in uence the behaviour. This phenomenon has important implications for the design of robust
algorithms. The choice of error tolerance, however, is shown not to a ect the stability of the equilibrium states.
Numerical tests are used to illustrate the analysis. Finally, a general result is given that guarantees the existence
of equilibrium states for a large class of algorithms.
Title: The Dynamics of a Discretized Nonlinear Delay Di erential Equation
Authors: D J Higham
Reference: Tech. Rep. NA/148 (1993), Dept. Math. Comp. Sci., Univ. Dundee.
Abstract: We consider a delay-logistic di erential equation that has important applications in population dy-
namics. We set up a constant-stepsize discretisation of the problem and examine the long term behaviour of the
solution. Key di erences between this problem and the more familiar non-delay version are that
1. introducing a small delay improves the stability of the iteration,
2. for a certain parameter range, the stable steady state loses stability through a complex eigenvalue, and
3. spurious solutions exist even when the stepsize is stable in a linear sense.
We also consider a variable-stepsize version of the discrete method, based on a standard local error estimate. Tests
reveal that the error control suppresses spurious behaviour, and we give some theoretical arguments to support
these observations.
Title: The Stability of a Class of Runge-Kutta Methods for Delay Di erential Equations
Authors: K J in't Hout
Abstractz: This paper deals with the stability analysis of Runge-Kutta methods for delay di erential equations.
We focus on the subclass of collocation methods with abscissae in [0;1), and we prove that all of these methods
violate an important stability condition related to a class of test problems U0(t) = U(t)+U(t?) with ; 2 C,
() ?jj, and 0.
Title: Stability Analysis of Numerical Methods for Delay Di erential Equations
Authors: K J in't Hout and M N Spijker
Abstract: This paper deals with the stability analysis of step-by-step methods for the numerical solution of delay
di erential equations. We focus on the behaviour of such methods when they are applied to the linear test problem
U0(t) = U(t) + U(t ? ) with 0 and ; complex. A general theorem is presented which can be used to
obtain complete characterizations of the stability regions of these methods.
Title: Improved Absolute Stability of Predictor-Corrector Methods for Retarded Di erential Equations
Authors: P J van der Houwen and B P Sommeijer
Reference: Di erential-Di erence Equations, ISNM 62, 137{148, Birkhauser 1983.
Reference: Tech. Rep. 135 (1982), Dept. Numer. Math., CWI.
23

Abstractz: The absolute stability of predictor-corrector type methods is investigated for retarded di erential
equations. The stability test equation is of the form y0(t) = !1y(t) + !2y(t ? !) where !1, !2 and ! are constants
(! 0). By generalizing the conventional predictor-corrector methods it is possible to improve the stability region
in the (!1t;!2t)-plane considerably. In particular, methods based on extrapolation-predictors and backward
di erentiation-correctors are studied.
Title: Stability in Linear Multistep Methods for Pure Delay Equations
Authors: P J van der Houwen and B P Sommeijer
Reference: Tech. Rep. 152 (1983), Dept. Numer. Math., CWI.
Abstractz: The stability regions of linear multistep methods for pure delay equations are compared with the
stabilityregion ofthe delay equationitself. A criterion is derived stating when the numericalstabilityregioncontains
the analyticalstabilityregion. This criterion yields an upper bound for the integrationstep (conditionalQ-stability).
These bounds are computed for the Adams-Bashforth, Adams-Moulton and backward di erentiation methods of
order 8. Furthermore, symmetric Adamsmethods are considered which are shown to be unconditionallyQ-stable.
Finally, the extended backward di erentiation methods of Cash are analysed.
Title: On the Stability of Predictor-Corrector Methods for Parabolic Equations with Delay
Authors: P J van der Houwen, B P Sommeijer and C T H Baker
Abstract: Di usion problems where the current state depends upon an earlier one give rise to parabolic equations
with delay. The ecient numerical solution of classical parabolic equations can be accomplished via methods for
sti di erential equations; one such class is predictor-corrector-type methods with extended real stability intervals
and with reduced storage requirements. Analogous methods for equations with delay are proposed and analysed
here. Our analysis will be based on the test equation y0(t) = q1y(t) + q2y(t ? !), where, in view of the class of
parabolic delay equations we want to consider, our main interest will be in the case jq1j jq2j. Implementational
details of the methods developed are given and numerical results are presented.
Title: Asymptotic Stability Analysis of -Methods for Functional Di erential Equations
Authors: Z Jackiewicz
Abstract: Stability analysis of -methods for functional di erential equations based on the test equation
y0(t) = ay(t ?) + by(t); t 0;
y(t) = (t); t 2 [?;0];
0, is presented. It is known that y(t) ! 0 as t ! 1 if and only if jbj ?a and we investigate whether this
property is inherited by the numerical solution approximating y.
Title: The Functional Di erential Equation y0(x) = ay(x) + by(x)
Authors: T Kato and J B McLeod
Reference: Bull. Amer. Math. Soc., 77 (1971), 891{937.
Abstract: The paper discusses the functional di erential equation
y0(x) = ay(x) + by(x) (0 1); (3)
where a is a possible complex constant, b a real constant, and a non-negative constant.
The paper rst shows that the boundary-value problem associated with (3) and the boundary condition y(0) = 1 is
well-posed if 1, but not if 1.
24

The remainder of the paper discusses the asymptotic properties of solutions of the equation as x ! 1. If 1,
it is possible to discuss the asymptotics of all solutions of the equation; if 1, it is shown that, given a speci c
asymptotic behaviour, there is one and only one solution which possesses that asymptotic behaviour.
Title: Stability of some Test Equations with Delay
Authors: V B Kolmanosvskii, L Torelli and R Vermiglio
Reference: SIAM J. Math. Anal., 25 (1994), 948{961.
Reference: Tech. Rep. 261 (1992), Dept. Math. Sci., Trieste.
Abstractz: In this paper the authors propose some techniques to obtain stability conditions for certain di erential
equations with delay. These techniques are applied to three concrete test situations. In the rst and second cases
they consider equations without instantaneous dissipative terms. This is important because in real situations, for
example in control theory, nite time is necessary to measure characteristic of the object and to treat the result of
the measurement in order to create control action. Moreover, in the last section, they present general results for
the chemostat model. All these test equations are interesting examples for a next investigation concerning stability
of numerical methods.
Title: Oscillations and Global Attractivity in a Discrete Delay Logistic Model
Authors: S A Kuruklis and G Ladas
Reference: Quart. Appl. Math., 50 (1992), 227{233.
Abstract: Consider the discrete delay logistic equation
Nt+1 = Nt
1+ Nt?k
; (4)
where 2 (1;1), 2 (0;1), and k 2 N = f0;1;2;...g. We obtain conditions for the oscillation and asymptotic
stability of all positive solution of Eq. (4) about its positive equilibrium ( ? 1)= . We prove that all positive
solutions of Eq. (4) are bounded and that for k = 0 and k = 1 the positive equilibrium ( ? 1)= is a global
attractor.
Title: Oscillations of Nonlinear Functional Di erential Equations generated by Retarded Actions
Authors: G S Ladde
1972.
Abstract: Recently, an attempt was made by Ladas and Lakshmikantham[3], Ladas, Ladde and Papadakis [4] to
establish the oscillatory behaviour of linear second order retarded di erential equations generated by delays.
Inthis paper, our aimis to investigatethe oscillatoryand asymptoticbehaviour ofsolutionsofnonlinear second order
functional di erential equations. First, we classify all solutions of such equations with respect to their behaviour
as t ! 1 and to their oscillatory nature. Our results generalize the results in [4,5] for nonlinear second order
functional di erential equations. Finally, we extend the oscillation results to nonlinear second order functional
di erential equations under certain growth conditions on an equation and on delay. This oscillatory behaviour is
generated by retarded actions and vanishes when retarded action vanishes. Our work includes the earlier work [3,4]
as special cases.
Title: On a Nonlinear Delay Equation
Authors: J J Levin and J A Nohel
Reference: J. Math. Anal. Applic., 8 (1964), 31{44.
Abstract: We investigate the behaviour, as t ! 1, of the solutions of
x0(t) = ?1
L
Z t
t?L
(L? (t? ))g(x())d
25

where L 0 is a given constant and g(x) is the restoring force of a given spring, which is not necessarily linear.
Title: The Stability of -Methods in the Numerical Solution of Delay Di erential Equations
Authors: M Z Liu and M N Spijker
Abstractz: This paper deals with the stability analysis of numerical methods for the solution of delay di erential
equations. We focus on the behaviour of the one-leg -method and the linear -method in the solution of the linear
test equation U0(t) = U(t)+U(t?), with 0 and complex ;. The stability regions for both of the methods
are determined. The regions turn out to be equal to each other only if = 0 or = 1.
Title: The Stability of the Block -Methods
Authors: L Lu
Abstract: In this paper, the author proposes a class of block -methods for the numerical solution of ODEs and
DDEs. It is shown that the spectrum of the coecient matrix contains a single point. This enables us to reduce
the computing counts. The necessary and sucient conditions for A( ), A, L( ), L and P-stabilities are studied
in detail.
Title: A Test for Stability of Linear Di erential Delay Equations
Authors: J M Maha y
Reference: Quart. Appl. Math., (1982), 193{202.
Abstract: The changes in the stability of a system of linear di erential delay equations resulting from the delay
are studied by analyzing the associated eigenvalues of the characteristic equation. A speci c contour is mapped
by the characteristic equation into the complex plane to give an easy test for stability from an application of the
argument principle. When the real part of an eigenvalue is positive, the contour gives bounds on the imaginary
part which are important in certain applications to nonlinear problems.
Title: Asymptotic Behaviour of Nonlinear Delay Di erential Equations
Authors: R K Miller
Reference: J. Di . Eqn., 1 (1965), 293{305.
Abstract: In this paper we study the behaviour as t ! 1 of solutions of certain non-linear delay di erential
equations. We extend the notion of an invariant set to almost periodic equations and consider the interaction of
this notion of invariance with perturbation theory and with the theory of Liapunov functions.
Title: Stability of x0(t) = Ax(t)+ Bx(t? )
Authors: T Mori and H Kokame
Reference: IEEE Trans. Autom. Control, 34 (1989), 460{462.
Abstract: A stability criterion for linear time-delay systems described by a di erential equation of the form
x0(t) = Ax(t) + Bx(t ? ) is proposed. The result obtained can include information on the size of the delay, and
therefore can be a delay-dependent stability condition. Its relation to existing delay-independent stability criterion
is also discussed.
Title: Characterization of Jump Discontinuities for State Dependent Delay Di erential Equations
Authors: K W Neves and A Feldstein
Reference: J. Math. Anal. Applic., 56 (1976), 689{707.
Abstractz: A number of authors have posed applications that lead to functional di erential equations (FDEs)
26

with state dependent delays (that is, delays which depend upon the unknown solution). For instance Cooke has
pointed out that population models and infection models lead, in simple cases, to the equation
u0(t) +au(t? r(u(t))) = 0
where r(u) 0, r(0) = 0, a 0. Cooke discussed the appropriateness of this model, and he presented some
asymptotic theory. As early as 1960 Driver pointed out that the two body problem of classical electrodynamics is
properly modelled as an FDE with state dependent delays. Furthermore, he was (to the best of our knowledge) the
rst author to present an existence and uniqueness theory for such equations.
The growth of interest in FDEs has been stimulated by their wide-spread applications. FDEs of various forms
provide the basis for mathematical models in numerous areas, including neural network theory, theory of learning,
epidemiology, pharmacokinetics and time lag control process. For more detailed study of applications of FDEs see
the book edited by Schmitt.
In this paper we shall consider the scalar equation
x0(t) = f(t;x(t);x( (t;x(t)))) for t 2 [a;b]
with x(t) = (t) for t 2 [
a;a] where
a = min (t;x(t)) for t 2 [a;b].
Title: Characterization of Periodic Solutions of Special Di erential Delay Equations
Authors: D Saupe
Reference: Lect. Notes Math., 1017 (1983), 553{562.
Reference: Tech. Rep. 72 (1982), Forsch. Dynam. Syst., Univ. Bremen.
Abstract: The aim of this paper is to derive a simple nite dimensional characterization of periodic solutions of
the di erential delay equation
x0(t) = ?f(x(t ? 1)); 0 (5)
where f : R ! R is an odd and piecewise constant function satisfying xf(x) 0 for all x 6= 0. Here a solution x of
(5) is a continuous piecewise linear function which solves the integrated version
x(t) = x(0)?
Z t?1
?1
f(x(s))ds
of (5). From our results we obtain a suitable numerical procedure for the computation of periodic solutions of (5)
and we report its performance on a test example.
Title: A Sucient Condition for GPN-stability for Delay Di erential Equations
Authors: L Torelli
Reference: Tech. Rep. 187 (1989), Dipart. Sci. Math., Univ. Trieste.
Abstractz: In this paper the author considers a linear test delay di erential equation with non-constant coecients
related to the de nition of PN and GPN-stability for numerical methods. He de nes stability properties for an
ordinary di erential equation together with stability properties of interpolants for numerical methods and in this
way he gives sucient conditions for GPN-stability.
Title: Stability of Numerical methods for Delay Di erential Equations
Authors: L Torelli
Abstract: Consider the following delay di erential equation (DDE)
y0(t) = f(t;y(t);y(t ? (t)); t t0 (6)
27

with the initial condition
y(t) = (t); t t0; (7)
where f and are such that (6) and (7) has a unique solution y(t). The author gives sucient conditions for the
asymptotic stability of the equation (6) for which he introduces new de nitions of numerical stability. The approach
is reminiscent of that from the nonlinear, sti ordinary di erential equation (ODE) eld. He investigates stability
properties of the class of one-point collocation rules. In particular, the backward Euler method turns out to be
stable with respect to all the given De nitions.
Title: On the Stability of Continuous Quadrature Rules for Di erential Equations with Several Constant Delays
Authors: L Torelli and R Vermiglio
Abstract: The aim of the present paper is to study the stability properties of the numerical methods for pure
delay di erential equations. The methods we consider are based on a quadrature rule and an interpolant (NCE) to
get an approximation of the retarded part (continuous quadrature rule). As a test equation we consider
y0(t) = ?
R
X
r=1
br(t)y(t ?r); t 0; y(t) = (t); t 0
and we give sucient conditions for the boundedness of the solutions. The same behaviour is preserved by the
continuous quadrature rule under some restriction on the parameters. As a conclusion we give some examples.
Title: Asymptotic Stability of a Time Delayed Di usion System
Authors: P K C Wang
Reference: Trans. ASME J. Appl. Mech., (1963), 500{504.
Abstract: This paper discusses the asymptotic stability of the equilibrium states of a nonlinear di usion system
with time delays. It is assumed that the system is describable by a partial di erential-di erence equation of the
form:
@u(t;X)
@t = Lu(t;X)+ f(t;X;u(t;X);u(t?T;X);...; @u(t;X)
@xi
;...; @u(t ?T;X)
@xi
;...) i = 1;...;M
where L is a linear operator uniformly elliptic in X de ned for all X 2 ... a bounded, open M-dimensional
spatial domain; f is a speci ed function of its arguments. In the development of this paper, the physical origin
of the foregoing equation is discussed brie y. Then, conditions for asymptotic stability of the trivial solution are
derived via an extended Lyapunov's direct method. Speci c results are given for a simple one-dimensional linear
heat equation with time-delayed arguments.
Title: Optimal Control of Parabolic Systems with Boundary Conditions involving Time Delays
Authors: P K C Wang
Reference: SIAM J. Control, 13 (1975), 274{293.
Abstract: Various optimal control problems with quadratic cost functionals for parabolic systems with Neumann
boundary conditions involving time delays are considered. Necessary and sucient conditions which the optimal
controls must satisfy are derived. Estimates and a sucient condition for the boundedness of solutions are obtained
for systems with speci ed forms of feedback controls. Similar problems for systems with more complex boundary
conditions involving time delays are discussed brie y.
Title: The Stability of Di erence Formulae for Delay Di erential Equations
Authors: D S Watanabe and M G Roth
Abstract: A new simple geometric technique is presented for analyzing the stability of di erence formula for the
28

model delay di erential equation
y0(t) = py(t) + qy(t ?);
where p and q are complex constants, and the delay is a positive constant. The technique is based on the argument
principle and directly relates the region of absolute stability for ordinary di erential equations corresponding to
the py(t) term with the region corresponding to the delay term qy(t ? ). A sucient condition for stability is
that these regions are disjoint. The technique is used to show that for each A-stable, A( )-stable, or stiy stable
linear multistep formula for ordinary di erential equations, there is a corresponding linear multistep formula for
delay di erential equations with analogous stability properties. The analogy does not extend, however, to A-stable
one-step formulae.
Title: Stability of Multistep Methods for Delay Di erential Equations
Authors: L F Wiederholt
Abstract: The absolute and relative stability of linear multistep methods for a nite stepsize is studied for delay
di erential equations. The di erential equations are assumed linear and the delays a constant integer multiples of
the stepsize. Computableconditions for stability are developed for scalar equations. Plots of the stability regions for
several common multistep methods are included. For the integration methods considered, the stability regions for
delay di erential equations are signi cantly di erent from the stability regions for ordinary di erential equations.
Title: On a Sequence de ned by a Non-Linear Recurrence Formula
Authors: E M Wright
Reference: J. London Math. Soc., 20 (1945), 68{73.
Abstract: In a recent series of articles, I have developed parts of the general theory of linear and non-linear
di erence-di erential equations. The signi cance and usefulness of the general results are somewhat obscured by
the comparative lack of information about speci c examples, such as is readily available in the special cases of
di erential and di erence equations. For this reason I discuss here the particular non-linear equation
y0(x) = ? y(x ?1)f1+ y(x)g ( 0): (8)
The general results already proved and various methods which I develop here enable me to deduce unexpectedly
detailed information about the solutions of (8), but there are still certain unsolved problems. The methods can be
applied more widely, but my treatment of (8) illustrates the general ideas suciently.
Title: The Non-Linear Di erence-Di erential Equation
Authors: E M Wright
Reference: Quart. J. Math., 17 (1946), 245{252.
Abstract: The general non-linear di erence-di erential equation with constant coecients may be written in the
form
1fy(x)g+ 2fy(x)g = v(x); (9)
where
1fy(x)g Pm
=0
Pn
=0ay()(x +b);
2fy(x)g P
Ay( ;1)(x+ b;1)y( ;2)(x+b;2)...;
y(0)(x) y(x);
and v(x) is a known function of the real variable x. We suppose that m 1, n 1 and 0 ;i n. The numbers
a, b, A are independent of x, and the b are real. 2fy(x)g has a nite number of terms and each term has at least
two y-function factors.
Particular examples of (9) have occurred in arithmetic, in the problem of the iterated exponential function and
29

A Bibliography on the Numerical Solution of Delay Differential Equations.pdf

A Bibliography on the Numerical Solution of Delay Differential Equations.pdf

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A Bibliography on the Numerical Solution of Delay Differential Equations.pdf