Arino o hbid m l ait dads e (eds ) delay differential equations and applications

Delay Differential Equations and Applications


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Series II: Mathematics, Physics and Chemistry – Vol. 205


Delay Differential Equations
and Applications
edited by

O. Arino
University of Pau, France

M.L. Hbid
University Cadi Ayyad,
Marrakech, Morocco
and

E. Ait Dads
University Cadi Ayyad,
Marrakech, Morocco

Published in cooperation with NATO Public Diplomacy Division


Proceedings of the NATO Advanced Study Institute on
Delay Differential Equations and Applications
Marrakech, Morocco

9–21 September 2002
A C.I.P. Catalogue record for this book is available from the Library of Congress.

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ISBN-13
ISBN-10
ISBN-13
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1-4020-3646-9 (PB)
978-1-4020-3646-0 (PB)
1-4020-3645-0 (HB)
978-1-4020-3645-3 (HB)
1-4020-3647-7 (e-book)
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Published by Springer,
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Contents

List of Figures

xiii

Preface

xvii

Contributing Authors

xxi

Introduction
M. L. Hbid

xxiii

1
History Of Delay Equations
J.K. Hale
1
Stability of equilibria and Lyapunov functions
2
Invariant Sets, Omega-limits and Lyapunov functionals.
3

Delays may cause instability.
4
Linear autonomous equations and perturbations.
5
Neutral Functional Differential Equations
6
Periodically forced systems and discrete dynamical systems.
7
Dissipation, maximal compact invariant sets and attractors.
8
Stationary points of dissipative flows
Part I General Results and Linear Theory of Delay Equations in
Finite Dimensional Spaces
2
Some General Results and Remarks on Delay Differential Equations
E. Ait Dads
1
Introduction
2
A general initial value problem
2.1
Existence
2.2
Uniqueness
2.3
Continuation of solutions
2.4
Dependence on initial values and parameters
2.5
Differentiability of solutions

v

1
3
7
10
12
16
20
21
24

29
31
31
33
34
35
37
38
40


vi

DELAY DIFFERENTIAL EQUATIONS

3
Autonomous Functional Differential Equations

Franz Kappel
1
Basic Theory
1.1
Preliminaries
1.2
Existence and uniqueness of solutions
1.3
The Laplace-transform of solutions. The fundamental
matrix
1.4
Smooth initial functions
1.5
The variation of constants formula
1.6
The Spectrum
1.7
The solution semigroup
2
Eigenspaces
2.1
Generalized eigenspaces
2.2
Projections onto eigenspaces
2.3
Exponential dichotomy of the state space
3
Small Solutions and Completeness
3.1
Small solutions

3.2
Completeness of generalized eigenfunctions
4
Degenerate delay equations
4.1
A necessary and sufficient condition
4.2
A necessary condition for degeneracy
4.3
Coordinate transformations with delays
4.4
The structure of degenerate systems
with commensurate delays
Appendix: A
Appendix: B
Appendix: C
Appendix: D

46
54
55
59
68
71
71
90
101
104
104
109

110
110
116
119

References

137

Part II Hopf Bifurcation, Centre manifolds and Normal Forms for Delay Differential Equations
4
Variation of Constant Formula for Delay Differential Equations
M.L. Hbid and K. Ezzinbi
1
Introduction
2

3

Variation Of Constant Formula Using Sun-Star Machinery
2.1
Duality and semigroups
2.1.1 The variation of constant formula:
2.2
Application to delay differential equations
2.2.1 The trivial equation:
2.2.2 The general equation
Variation Of Constant Formula Using Integrated Semigroups
Theory
3.1

Notations and basic results
3.2
The variation of constant formula

41
41
41
44

124
127
129
131
132

141
143
143
145
145
146
147
147
149
149
150
153


Contents

5
Introduction to Hopf Bifurcation Theory for Delay Differential
Equations
M.L. Hbid
1
Introduction
1.1
Statement of the Problem:
1.2
History of the problem
1.2.1 The Case of ODEs:
1.2.2 The case of Delay Equations:
2
The Lyapunov Direct Method And Hopf Bifurcation: The Case
Of Ode
3
The Center Manifold Reduction Of DDE
3.1
The linear equation
3.2
The center manifold theorem
3.3
Back to the nonlinear equation:
3.4
The reduced system
4
Cases Where The Approximation Of Center Manifold Is
Needed
4.1
Approximation of a local center manifold

4.2
The reduced system
6
An Algorithmic Scheme for Approximating Center Manifolds
and Normal Forms for Functional Differential Equations
M. Ait Babram
1
Introduction
2
Notations and background
3
Computational scheme of a local center manifold
3.1
Formulation of the scheme
3.2
Special cases.
3.2.1 Case of Hopf singularity
3.2.2 The case of Bogdanov -Takens singularity.
4
Computational scheme of Normal Forms
4.1
Normal form construction of the reduced system
4.2
Normal form construction for FDEs
7
Normal Forms and Bifurcations for Delay Differential Equations
T. Faria
1
Introduction
2

Normal Forms for FDEs in Finite Dimensional Spaces
2.1
Preliminaries
2.2
The enlarged phase space
2.3
Normal form construction
2.4
Equations with parameters
2.5
More about normal forms for FDEs in Rn
3
Normal forms and Bifurcation Problems
3.1
The Bogdanov-Takens bifurcation
3.2
Hopf bifurcation

vii
161
161
161
163
163
164
166
168
169
172
177

179
182
183
188
193
193
195
199
202
209
209
210
213
214
221

227
227
231
231
232
234
240
241
243
243
246


viii

DELAY DIFFERENTIAL EQUATIONS
4

5

Normal Forms for FDEs in Hilbert Spaces
4.1
Linear FDEs
4.2
Normal forms
4.3
The associated FDE on R
4.4
Applications to bifurcation problems
Normal Forms for FDEs in General Banach Spaces
5.1
Adjoint theory
5.2
Normal forms on centre manifolds
5.3
A reaction-diffusion equation with delay and Dirichlet
conditions

References
Part III Functional Differential Equations in Infinite Dimensional
Spaces

253
254

256
258
260
262
263
268
270
275

283

8
285
A Theory of Linear Delay Differential Equations in Infinite Dimensional Spaces
O. Arino and E. S´
anchez
1
Introduction
285
1.1
A model of fish population dynamics with spatial diffusion (11)
286
1.2
An abstract differential equation arising from cell population dynamics
288
1.3
From integro-difference to abstract delay differential equations (8)
292
1.3.1 The linear equation
292

1.3.2 Delay differential equation formulation of system (1.5)(1.6)
295
1.4
The linearized equation of equation (1.17) near nontrivial steady-states
297
1.4.1 The steady-state equation
297
298
1.4.2 Linearization of equation (1.17 ) near (n, N )
1.4.3 Exponential solutions of (1.20)
299
1.5
Conclusion
303
2
The Cauchy Problem For An Abstract Linear Delay Differential Equation
303
2.1
Resolution of the Cauchy problem
304
2.2
Semigroup approach to the problem (CP)
306
310
2.3
Some results about the range of λI − A
3
Formal Duality
311
3.1

The formal adjoint equation
313
316
3.2
The operator A∗ formal adjoint of A
3.3
Application to the model of cell population dynamics
317
3.4
Conclusion
320
4
Linear Theory Of Abstract Functional Differential Equations
Of Retarded Type
320
321
4.1
Some spectral properties of C0 -semigroups


ix

Contents
4.2
4.3
4.4

Decomposition of the state space C([−r, 0]; E)
A Fredholm alternative principle
Characterization of the subspace R (λI − A)m for λ in

324
326

(σ\σe ) (A)

326

4.5

5

Characterization of the projection operator onto the
subspace QΛ
4.6
Conclusion
A Variation Of Constants Formula For An Abstract Functional
Differential Equation Of Retarded Type
5.1
The nonhomogeneous problem
5.2
Semigroup defined in L(E)
5.3
The fundamental solution
5.4
The fundamental solution and the nonhomogeneous
problem
5.5
Decomposition of the nonhomogeneous problem
in C([−r, 0]; E)

9
The Basic Theory of Abstract Semilinear Functional Differential
Equations with Non-Dense Domain
K. Ezzinbi and M. Adimy
1
Introduction
2
Basic results
3
Existence, uniqueness and regularity of solutions
4
The semigroup and the integrated semigroup in the autonomous
case
5
Principle of linearized stability
6
Spectral Decomposition
7
Existence of bounded solutions
8
Existence of periodic or almost periodic solutions
9
Applications
References
Part IV More on Delay Differential Equations and Applications
10
Dynamics of Delay Differential Equations
H.O. Walther
1

Basic theory and some results for examples
1.1
Semiflows of retarded functional differential equations
1.2
Periodic orbits and Poincar´e return maps
1.3
Compactness
1.4
Global attractors
1.5
Linear autonomous equations and spectral
decomposition
1.6
Local invariant manifolds for nonlinear RFDEs
1.7
Floquet multipliers of periodic orbits
1.8
Differential equations with state-dependent delays

331
335
335
336
337
338
341
344
347
347
350

354
372
381
383
385
391
393
399
409
411
411
411
416
418
418
419
423
425
435


x

DELAY DIFFERENTIAL EQUATIONS
2

3
4
5

Monotone feedback: The structure of invariant sets and
attractors
2.1
Negative feedback
2.2
Positive feedback
Chaotic motion
Stable periodic orbits
State-dependent delays

11
Delay Differential Equations in Single Species Dynamics
S. Ruan
1
Introduction
2
Hutchinson’s Equation
2.1
Stability and Bifurcation
2.2
Wright Conjecture
2.3
Instantaneous Dominance
3
Recruitment Models
3.1
Nicholson’s Blowflies Model
3.2
Houseflies Model
3.3

Recruitment Models
4
The Allee Effect
5
Food-Limited Models
6
Regulation of Haematopoiesis
6.1
Mackey-Glass Models
6.2
Wazewska-Czyzewska and Lasota Model
7
A Vector Disease Model
8
Multiple Delays
9
Volterra Integrodifferential Equations
9.1
Weak Kernel
9.2
Strong Kernel
9.3
General Kernel
9.4
Remarks
10
Periodicity
10.1 Periodic Delay Models
10.2 Integrodifferential Equations
11

State-Dependent Delays
12
Diffusive Models with Delay
12.1 Fisher Equation
12.2 Diffusive Equations with Delay
12
Well-Posedness, Regularity and Asymptotic Behaviour of Retarded Differential Equations by Extrapolation Theory
L. Maniar
1
Introduction
2
Preliminaries
3
Homogeneous Retarded Differential Equations

436
437
439
451
456
468
477
477
478
479
481
483
484
484
486

487
488
489
491
491
493
493
495
496
498
500
502
504
505
505
507
511
514
514
515
519
519
521
525


Contents

xi

13
Time Delays in Epidemic Models: Modeling and Numerical Considerations
J. Arino and P. van den Driessche
1
Introduction
2
Origin of time delays in epidemic models
2.1
Sojourn times and survival functions
2.2
Sojourn times in an SIS disease transmission model
3
A model that includes a vaccinated state
4
Reduction of the system by using specific P (t) functions
4.1
Case reducing to an ODE system
4.2
Case reducing to a delay integro-differential system
5
Numerical considerations
5.1
Visualising and locating the bifurcation
5.2
Numerical bifurcation analysis and integration
6
A few words of warning
Appendix
1
Program listings

1.1
MatLab code
1.2
XPPAUT code
2
Delay differential equations packages
2.1
Numerical integration
2.2
Bifurcation analysis

539
540
540
541
544
548
548
549
550
550
551
552
555
555
556
557
557
557
558

References

559

Index

579

539


List of Figures

10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
10.13
10.14
10.15
10.16

10.17
10.18
10.19
10.20
10.21
10.22
10.23
10.24
10.25
10.26
10.27
10.28
10.29

413
413
417
417
420
422
423
424
425
427
428
430
431
432
432
434

438
439
441
442
443
445
446
447
447
449
450
451
453
xiii


xiv

DELAY DIFFERENTIAL EQUATIONS

10.30

453

10.31

454

10.32

457

10.33

459

10.34

459

10.35

463

10.36

464

10.37

465

10.38

466

11.1

The bifurcation diagram for equation (2.1).

481

11.2

The periodic solution of the Hutchinson’s equation
(2.1).

481

Numerical simulations for the Hutchinson’s equation (2.1). Here r = 0.15, K = 1.00. (i) When τ = 8,
the steady state x∗ = 1 is stable; (ii) When τ = 11,
a periodic solution bifurcated from x∗ = 1.

482

Oscillations in the Nicholson’s blowflies equation (3.1).
Here P = 8, x0 = 4, δ = 0.175, and τ = 15.

486

Aperiodic oscillations in the Nicholson’s blowflies
equation (3.1). Here P = 8, x0 = 4, δ = 0.475, and
τ = 15.

486

Numerical simulations in the houseflies model (3.2).
Here the parameter values b = 1.81, k = 0.5107, d =
0.147, z = 0.000226, τ = 5 were reported in Taylor
and Sokal (1976).

487

The steady state of the delay model (4.1) is attractive. Here a = 1, b = 1, c = 0.5, τ = 0.2.

489

The steady state of the delay food-limited model
(5.3) is stable for small delay (τ = 8) and unstable
for large delay (τ = 12.8). Here r = 0.15, K =
1.00, c = 1.

491

Oscillations in the Mackey-Glass model (6.1). Here
λ = 0.2, a = 01, g = 0.1, m = 10 and τ = 6.

492

11.3

11.4
11.5

11.6

11.7
11.8

11.9

11.10 Aperiodic behavior of the solutions of the MackeyGlass model (6.1). Here λ = 0.2, a = 01, g = 0.1, m =
10 and τ = 20.

492


List of Figures

11.11 Numerical simulations for the vector disease equation (7.1). When a = 5.8, b = 4.8(a > b), the zero
steady state u = 0 is asymptotically stable; When
a = 3.8, b = 4.8(a < b), the positive steady state u∗
is asymptotically stable for all delay values; here for
both cases τ = 5.
11.12 For the two delay logistic model (8.1), choose r =
0.15, a1 = 0.25, a2 = 0.75. (a) The steady state (a)
is stable when τ1 = 15 and τ2 = 5 and (b) becomes
unstable when τ1 = 15 and τ2 = 10, a Hopf bifurcation occurs.
11.13 (a) Weak delay kernel and (b) strong delay kernel.
11.14 The steady state of the integrodifferential equation
(9.1) is globally stable. Here r = 0.15, K = 1.00.
11.15 The steady state x∗ = K of the integrodifferential
equation (9.10) losses stability and a Hopf bifurcation occurs when α changes from 0.65 to 0.065. Here
r = 0.15, K = 1.00.
11.16 Numerical simulations for the state-dependent delay
model (11.3) with r = 0.15, K = 1.00 and τ (x) =
a + bx2 . (i) a = 5, b = 1.1; and (ii) a = 9.1541, b = 1.1.
11.17 The traveling front profiles for the Fisher equation
(12.1). Here D = r = K = 1, c = 2.4 − 3.0
13.1 The transfer diagram for the SIS model.

13.2 The transfer diagram for the SIV model.
13.3 Possible bifurcation scenarios.
13.4 Bifurcation diagram and some solutions of (4.3). (a)
and (b): Backward bifurcation case, parameters as
in the text. (c) and (d): Forward bifurcation case,
parameters as in the text except that σ = 0.3.
13.5 Value of I ∗ as a function of ω by solving H(I, ω) = 0,
parameters as in text.
13.6 Plot of the solution of (6.2), with parameters as in
the text, using dde23.

xv

494

496
498
500

502

512
515
542
545
551

553
554
555

Preface

This book groups material that was used for the Marrakech 2002
School on Delay Differential Equations and Applications. The school
was held from September 9-21 2002 at the Semlalia College of Sciences
of the Cadi Ayyad University, Marrakech, Morocco. 47 participants and
15 instructors originating from 21 countries attended the school. Financial limitations only allowed support for part of the people from Africa
and Asia who had expressed their interest in the school and had hoped to
come. The school was supported by financements from NATO-ASI (Nato
advanced School), the International Centre of Pure and Applied Mathematics (CIMPA, Nice, France) and Cadi Ayyad University. The activity
of the school consisted in courses, plenary lectures (3) and communications (9), from Monday through Friday, 8.30 am to 6.30 pm. Courses
were divided into units of 45mn duration, taught by block of two units,
with a short 5mn break between two units within a block, and a 25mn
break between two blocks. The school was intended for mathematicians
willing to acquire some familiarity with delay differential equations or
enhance their knowledge on this subject. The aim was indeed to extend
the basic set of knowledge, including ordinary differential equations and
semilinear evolution equations, such as for example the diffusion-reaction
equations arising in morphogenesis or the Belouzov-Zhabotinsky chemical reaction, and the classic approach for the resolution of these equations by perturbation, to equations having in addition terms involving
past values of the solution. In order to achieve this goal, a training
programme was devised that may be summarized by the following three
keywords: the Cauchy problem, the variation of constants formula, local
study of equilibria. This defines the general method for the resolution of
semilinear evolution equations, such as the diffusion-reaction equation,
adapted to delay differential equations. The delay introduces specific
differences and difficulties which are taken into account in the progression of the course, the first week having been devoted to “ordinary”
delay differential equations, such equations where the only independent
variable is the time variable; in addition, only the finite dimension was

xvii


xviii

DELAY DIFFERENTIAL EQUATIONS

considered. During the second week, attention was focused on “ordinary” delay differential equations in infinite dimensional vector spaces,
as well as on partial differential equations with delay. Aside the training
on the basic theory of delay differential equations, the course by John
Mallet-Paret during the first week discussed very recent results motivated by the problem of determining wave fronts in lattice differential
equations. The problem gives rise to a differential equation with deviated arguments (both retarded and advanced), which represents an
entirely new line of research. Also, during the second week, Hans-Otto
Walther presented results regarding existence and description of the attractor of a scalar delay differential equation. Three plenary conferences
usefully extended the contents of the first week courses. The main part
of the courses given in the school are reproduced as lectures notes in
this book. A quick description of the contents the book is given in the
general introduction.
As many events of this nature at that time, this school was under
the scientific supervision of Ovide Arino. He wanted this book to be
published, and did a lot to that effect. He unfortunately passed away
on September 29, 2003. This book is dedicated to him.
J. Arino and M.L. Hbid


This book is dedicated
to the memory of
Professor Ovide Arino

Contributing Authors

Elhadi Ait Dads, Professor, Cadi Ayyad University, Marrakech, Morocco.
Mohammed Ait Babram, Assistant Professor, Cadi Ayyad University
of Marrakech, Morocco.
Mostafa Adimy, Professor, University of Pau, France.
Julien Arino, Assitant Professor, University of Manitoba, Winnipeg,
Manitoba, Canada.
Julien took over edition of the book after Ovide’s death.
Ovide Arino, Professor, Institut de Recherche pour le Developpement,
Centre de Bondy, and Universit´e de Pau, France.
P. van den Driessche, Professor, University of Victoria, Victoria,
British Columbia, Canada.
Khalil Ezzinbi, Professor, Cadi Ayyad University, Marrakech, Morocco.
Jack K. Hale, Professor Emeritus, Georgia Institute of Technology,
Atlanta, USA.
Tereza Faria, Professor, University of Lisboa, Portugal
Moulay Lhassan Hbid, Professor, Cadi Ayyad University, Marrakech,
Morocco.

xxi


xxii

DELAY DIFFERENTIAL EQUATIONS

Franz Kappel, Professor, University of Graz, Austria.
Lahcen Maniar, Professor, Cadi Ayyad University, Marrakech, Morocco.
Shigui Ruan, Professor, University of Miami, USA.

Eva Sanchez, Professor, University Polytecnica de Madrid, Spain.
Hans-Otto Walther, Professor, University of Giessen, Germany.
Said Boulite, Post-Doctoral Fellow, Cadi Ayyad University, Marrakech,
Morocco. (Technical realization of the book).


Introduction
M. L. Hbid

This book is devoted to the theory of delay equations and applications. It consists of four parts, preceded by an overview by Professor
J.K. Hale. The first part concerns some general results on the quantitative aspects of non-linear delay differential equations, by Professor E.
Ait Dads, and a linear theory of delay differential equations (DDE) by
Professor F. Kappel. The second part deals with some qualitative theory of DDE : normal forms, centre manifold and Hopf bifurcation theory
in finite dimension. This part groups the contributions of Professor T.
Faria, Doctor M. Ait Babram and Professor M.L. Hbid. The third part
corresponds to the contributions of Professors O. Arino, E. Sanchez,
T. Faria, M. Adimy and K. Ezzinbi. It is devoted to discussions on
quantitative and qualitative aspects of functionnal differential equations
(FDE) in infinite dimension. The last part contains the contributions of
Professors H.O. Walther, S. Ruan, L. Maniar and J. Arino.
Ait Dads’s contribution deals with a direct method to provide an
existence result; he then derives a number of typical properties of DDE
and their solutions. An example of such exotic properties, discussed
in Ait Dads’s lectures, is the fact that, contrary to the flow associated
with a smooth ordinary differential system of equations, which is a local
diffeomorphism for all times, the semiflow associated with a DDE does
not extend backward in time, degenerates in finite time and can even
vanish in finite time. Many such properties are not yet understood and
would certainly deserve to be thoroughly investigated. The results and
conjectures presented by Ait Dads are classical and are for most of them

taken from a recent monograph by J. Hale and S. Verduyn Lunel on
the subject. Their inclusion in the initiation to DDE proposed by Ait
Dads is mainly intended to allow readers to get some familiarity with
the subject and open their horizons and possibly entice their appetite
for exploring new avenues.

xxiii


xxiv

DELAY DIFFERENTIAL EQUATIONS

In his lecture notes, Franz Kappel presents the construction of the elementary solution of a linear DDE using the Laplace transform. Even if it
is possible to proceed by direct methods, the Laplace transform provides
an explicit expression of the elementary solution, useful in the study of
spectral properties of DDEs. Kappel also dealt with a fundamental issue of the linear theory of delay differential equations, namely, that of
completeness, that is to say, when is the vector space spanned by the
eigenvectors total (dense in the state space)? This issue is tightly connected with another delicate and still open one, the existence of “small”
solutions (solutions which approach zero at infinity faster than any exponential). This course extends the one that Prof. Kappel taught during
the first school on delay differential equations held at the University of
Marrakech in 1995. The very complete and elaborate lecture notes he
provided for the course are in fact an extension of the ones written on
the occasion of the first school. A first application of the linear and
the semi linear theory presented by Ait Dads and Kappel is the study
of bifurcation of equilibria in nonlinear delay differential equations dependent on one or several parameters. The typical framework here is
a DDE defined in an open subset of the state space, rather a family
of such equations dependent upon one parameter, which possesses for
each value of the parameter a known equilibrium (the so-called “trivial
equilibrium”): one studies the stability of the equilibrium and the possible changes in the linear stability status and how these changes reflect

in the local dynamics of the nonlinear equation. Changes are expected
near values of the parameter for which the equilibrium is a center. The
delay introduces its own problems in that case, and these problems have
given rise to a variety of approaches, dependent on the nature of the
delay and, more recently, on the dimension of the underlying space of
trajectories.
The part undertaken jointly by M.L. Hbid and M. Ait Babram deals
with a panorama of the best known methods, then concentrates on a
method elaborated within the dynamical systems group at the Cadi
Ayyad University, that is, the direct Lyapunov method. This method
consists in looking for a Lyapunov function associated with the ordinary
differential equation obtained by restricting the DDE to a center manifold. The Lyapunov function is determined recursively in the form of a
Taylor expansion. The same issue, in the context of partial differential
equations with delay, was dealt with by T. Faria in her lecture notes.
The method presented by Faria is an extension to this infinite dimensional frame of the well-known method of normal forms. The method
was presented both in the case of a delay differential equation and also
in the case when the equation is the sum of a delay differential equation


INTRODUCTION

xxv

and a diffusion operator. Both Prof. Faria and Dr. Ait Babram discuss
the Bogdanov-Takens and the Hopf bifurcation singularities as examples, and give a generic scheme to approximate the center manifolds in
both cases of singularies (Hopf, Bogdanov-Takens, Hopf-Hopf, ..).
The lecture notes written by Professor Hans-Otto Walther are composed of two independent parts: the first part deals with the geometry
of the attractor of the dynamical system defined by a scalar delay differential equation with monotone feedback. Both a negative and a positive
feedback were envisaged by Walther and his coworkers. In collaboration
with Dr. Tibor Krisztin, from the University of Szeged, Hungary, and

Professor Jianhong Wu, Fields Institute, Toronto, Canada, very detailed
global results on the geometric nature of the attractor and the flow along
the attractor were found. These results have been obtained within the
past ten years or so and are presented in a number of articles and monographs, the last one being more than 200 pages long. The course could
only give a general idea of the general procedure that was followed in
proving those results and was mainly intended to elicit the interest of
participants. The second part of Walther’s lecture notes is devoted to a
presentation of very recent results obtained by Walther in the study of
state-dependent delay differential equations.
The lectures notes by Professors O. Arino, K. Ezzinbi and M. Adimy,
and L. Maniar present approaches along the line of the semigroup theory.
These lectures prolong in the framework of infinite dimension the presentations made during the first part by Ait Dads and Kappel in the case
of finite dimensions. Altogether, they constitute a state of the art of the
treatment of the Cauchy problem in the frame of linear functional differential equations. The equations under investigation range from delay
differential equations defined by a bounded “delay” operator to equations in which the “delay” operator has a domain which is only part of
a larger space (it may be for example the sum of the Laplace operator
and a bounded operator), to neutral type equations in which the delay
appears also in the time derivative, to infinite delay, both in the autonomous and the non autonomous cases. The methods presented range
from the classical theory of strongly continuous semigroups to extrapolation theory, also including the theory of integrated semigroups and
the theory of perturbation by duality. Adimy and Ezzinbi dealt with a
general neutral equation perturbed by the Laplace operator. Arino presented a theory, elaborated in collaboration with Professor Eva Sanchez,
which extends to infinite dimensions the classical linear theory, as it is
treated in the monograph by Hale and Lunel.
In his lecture notes, S. Ruan provides a thorough review of models
involving delays in ecology, pointing out the significance of the delay.


xxvi

DELAY DIFFERENTIAL EQUATIONS

Most of his concern is about stability, stability loss and the corresponding
changes in the dynamical features of the problem. The methods used by
Ruan are those developed by Faria and Magalhaes in a series of papers,
which have been extensively described by Faria in her lectures. Dr. J.
Arino discusses the issue of delay in models of epidemics.
Various aspects of the theory of delay differential equations are presented in this book, including the Cauchy problem, the linear theory in
finite and in infinite dimensions, semilinear equations. Various types of
functional differential equations are considered in addition to the usual
DDE: neutral delay equations, equations with delay dependent upon the
starter, DDE with infinite delay, stochastic DDE, etc. The methods of
resolution covered most of the currently known ones, starting from the
direct method, the semigroup approach, as well as the integrated semigroup or the so-called sun-star approach. The lecture notes touched a
variety of issues, including the geometry of the attractor, the Hopf and
Bogdanov-Takens singularities. All this however is just a small portion
of the theory of DDE. We might name many subjects which haven’t
been or have just been briefly mentioned in lectures notes: the second
Lyapunov method for the study of stability, the Lyapunov-Razumikin
method briefly alluded to in the introductory lectures by Hale, the theory
of monotone (with respect to an order relation) semi flows for DDE which
plays an important role in applications to ecology (cooperative systems)
was considered only in the scalar case (the equation with positive feedback in Walther’s course). The prolific theory of oscillations for DDE
was not even mentioned, nor the DDE with impulses which are an important example in applications. The Morse decomposition, just briefly
reviewed the “structural stability” approach, of fundamental importance
in applications where it notably justifies robustness of model representations, a breakthrough accomplished during the 1985-1995 decade by
Mallet-Paret and coworkers is just mentioned in Walther’s course. Delay
differential equations have become a domain too wide for being covered
in just one book.

Chapter 1
HISTORY OF DELAY EQUATIONS
J.K. Hale
Georgia Institute of Techology
Atlanta, USA


Delay differential equations, differential integral equations and functional differential equations have been studied for at least 200 years (see
E. Schmitt (1911) for references and some properties of linear equations). Some of the early work originated from problems in geometry
and number theory.
At the international conference of mathematicians, Picard (1908) made
the following statement in which he emphasized the importance of the
consideration of hereditary effects in the modeling of physical systems:
Les ´equations diff´erentielles de la m´ecanique classique sont telles qu’il
en r´esulte que le mouvement est d´etermin´e par la simple connaissance
des positions et des vitesses, c’est-`
a-dire par l’´etat `
a un instant donn´e et
a l’instant infiniment voison.
`
Les ´etats ant´erieurs n’y intervenant pas, l’h´er´edit´e y est un vain mot.
L’application de ces ´equations o`
u le pass´e ne se distingue pas de l’avenir,
o`
u les mouvements sont de nature r´eversible, sont donc inapplicables aux
ˆetres vivants.
Nous pouvons r´ever d’´equations fonctionnelles plus compliqu´ees que
les ´equations classiques parce qu’elles renfermeront en outre des int´egrales
prises entre un temps pass´e tr`es ´eloign´e et le temps actuel, qui apporteront la part de l’h´er´edit´e.
Volterra (1909), (1928) discussed the integrodifferential equations that

model viscoelasticity. In (1931), he wrote a fundamental book on the
role of hereditary effects on models for the interaction of species.
The subject gained much momentum (especially in the Soviet Union)
after 1940 due to the consideration of meaningful models of engineering

1
O. Arino et al. (eds.), Delay Differential Equations and Applications, 1–28.
© 2006 Springer.


2

DELAY DIFFERENTIAL EQUATIONS

systems and control. It is probably true that most engineers were well
aware of the fact that hereditary effects occur in physical systems, but
this effect was often ignored because there was insufficient theory to
discuss such models in detail.
During the last 50 years, the theory of functional differential equations
has been developed extensively and has become part of the vocabulary
of researchers dealing with specific applications such as viscoelasticity,
mechanics, nuclear reactors, distributed networks, heat flow, neural networks, combustion, interaction of species, microbiology, learning models,
epidemiology, physiology,as well as many others (see Kolmanovski and
Myshkis (1999)).
Stochastic effects are also being considered but the theory is not as
well developed.
During the 1950’s, there was considerable activity in the subject which
led to important publications by Myshkis (1951), Krasovskii (1959), Bellman and Cooke (1963), Halanay (1966). These books give a clear picture
of the subject up to the early 1960’s.
Most research on functional differential equations (FDE) dealt primarily with linear equations and the preservation of stability (or instability)

of equilibria under small nonlinear perturbations when the linearization was stable (or unstable). For the linear equations with constant
coefficients, it was natural to use the Laplace transform. This led to expansions of solutions in terms of the eigenfunctions and the convergence
properties of these expansions.
For the stability of equilibria, it was important to understand the
extent to which one could apply the second method of Lyapunov (1891).
The genesis of the modern theory evolved from the consideration of the
latter problem.
In these lectures, I describe a few problems for which the method of
solution, in my opinion, played a very important role in the modern
analytic and geometric theory of FDE. At the present time, much of the
subject can be considered as well developed as the corresponding one for
ordinary differential equations (ODE). Naturally, the topics chosen are
subjective and another person might have chosen completely different
ones.
It took considerable time to take an idea from ODE and to find the
appropriate way to express this idea in FDE. With our present knowledge of FDE, it is difficult not to wonder why most of the early papers
making connections between these two subjects were not written long
ago. However, the mode of thought on FDE at the time was contrary to
the new approach and sometimes not easily accepted. A new approach