Table Of ContentDelay 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.
ISBN-10 1-4020-3646-9 (PB)
ISBN-13 978-1-4020-3646-0 (PB)
ISBN-10 1-4020-3645-0 (HB)
ISBN-13 978-1-4020-3645-3 (HB)
ISBN-10 1-4020-3647-7 (e-book)
ISBN-13 978-1-4020-3647-7 (e-book)
Published by Springer,
P.O.Box 17, 3300 AADordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
All Rights Reserved
© 2006 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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Contents
List of Figures xiii
Preface xvii
Contributing Authors xxi
Introduction xxiii
M. L. Hbid
1
History Of Delay Equations 1
J.K. Hale
1 Stability of equilibria and Lyapunov functions 3
2 Invariant Sets, Omega-limits and Lyapunov functionals. 7
3 Delays may cause instability. 10
4 Linear autonomous equations and perturbations. 12
5 Neutral Functional Differential Equations 16
6 Periodically forced systems and discrete dynamical systems. 20
7 Dissipation, maximal compact invariant sets and attractors. 21
8 Stationary points of dissipative flows 24
Part I General Results and Linear Theory of Delay Equations in
Finite Dimensional Spaces 29
2
SomeGeneralResultsandRemarksonDelayDifferentialEquations 31
E. Ait Dads
1 Introduction 31
2 A general initial value problem 33
2.1 Existence 34
2.2 Uniqueness 35
2.3 Continuation of solutions 37
2.4 Dependence on initial values and parameters 38
2.5 Differentiability of solutions 40
v
vi DELAY DIFFERENTIAL EQUATIONS
3
Autonomous Functional Differential Equations 41
Franz Kappel
1 Basic Theory 41
1.1 Preliminaries 41
1.2 Existence and uniqueness of solutions 44
1.3 The Laplace-transform of solutions. The fundamental
matrix 46
1.4 Smooth initial functions 54
1.5 The variation of constants formula 55
1.6 The Spectrum 59
1.7 The solution semigroup 68
2 Eigenspaces 71
2.1 Generalized eigenspaces 71
2.2 Projections onto eigenspaces 90
2.3 Exponential dichotomy of the state space 101
3 Small Solutions and Completeness 104
3.1 Small solutions 104
3.2 Completeness of generalized eigenfunctions 109
4 Degenerate delay equations 110
4.1 A necessary and sufficient condition 110
4.2 A necessary condition for degeneracy 116
4.3 Coordinate transformations with delays 119
4.4 The structure of degenerate systems
with commensurate delays 124
Appendix: A 127
Appendix: B 129
Appendix: C 131
Appendix: D 132
References 137
Part II Hopf Bifurcation, Centre manifolds and Normal Forms for De-
lay Differential Equations 141
4
Variation of Constant Formula for Delay Differential Equations 143
M.L. Hbid and K. Ezzinbi
1 Introduction 143
2 Variation Of Constant Formula Using Sun-Star Machinery 145
2.1 Duality and semigroups 145
2.1.1 The variation of constant formula: 146
2.2 Application to delay differential equations 147
2.2.1 The trivial equation: 147
2.2.2 The general equation 149
3 Variation Of Constant Formula Using Integrated Semigroups
Theory 149
3.1 Notations and basic results 150
3.2 The variation of constant formula 153
Contents vii
5
Introduction to Hopf Bifurcation Theory for Delay Differential 161
Equations
M.L. Hbid
1 Introduction 161
1.1 Statement of the Problem: 161
1.2 History of the problem 163
1.2.1 The Case of ODEs: 163
1.2.2 The case of Delay Equations: 164
2 TheLyapunovDirectMethodAndHopfBifurcation: TheCase
Of Ode 166
3 The Center Manifold Reduction Of DDE 168
3.1 The linear equation 169
3.2 The center manifold theorem 172
3.3 Back to the nonlinear equation: 177
3.4 The reduced system 179
4 Cases Where The Approximation Of Center Manifold Is
Needed 182
4.1 Approximation of a local center manifold 183
4.2 The reduced system 188
6
An Algorithmic Scheme for Approximating Center Manifolds 193
and Normal Forms for Functional Differential Equations
M. Ait Babram
1 Introduction 193
2 Notations and background 195
3 Computational scheme of a local center manifold 199
3.1 Formulation of the scheme 202
3.2 Special cases. 209
3.2.1 Case of Hopf singularity 209
3.2.2 The case of Bogdanov -Takens singularity. 210
4 Computational scheme of Normal Forms 213
4.1 Normal form construction of the reduced system 214
4.2 Normal form construction for FDEs 221
7
NormalFormsandBifurcationsforDelayDifferentialEquations 227
T. Faria
1 Introduction 227
2 Normal Forms for FDEs in Finite Dimensional Spaces 231
2.1 Preliminaries 231
2.2 The enlarged phase space 232
2.3 Normal form construction 234
2.4 Equations with parameters 240
2.5 More about normal forms for FDEs in Rn 241
3 Normal forms and Bifurcation Problems 243
3.1 The Bogdanov-Takens bifurcation 243
3.2 Hopf bifurcation 246
viii DELAY DIFFERENTIAL EQUATIONS
4 Normal Forms for FDEs in Hilbert Spaces 253
4.1 Linear FDEs 254
4.2 Normal forms 256
R
4.3 The associated FDE on 258
4.4 Applications to bifurcation problems 260
5 Normal Forms for FDEs in General Banach Spaces 262
5.1 Adjointtheory 263
5.2 Normal forms on centre manifolds 268
5.3 A reaction-diffusion equation with delayand Dirichlet
conditions 270
References 275
Part III Functional Differential Equations in Infinite Dimensional
Spaces 283
8
A Theory of Linear Delay Differential Equations in Infinite Di- 285
mensional Spaces
O. Arino and E. Sa´nchez
1 Introduction 285
1.1 A model of fish population dynamics with spatial dif-
fusion (11) 286
1.2 An abstract differential equation arising from cell pop-
ulation dynamics 288
1.3 Fromintegro-differencetoabstractdelaydifferentialequa-
tions (8) 292
1.3.1 The linear equation 292
1.3.2 Delaydifferentialequationformulationofsystem(1.5)-
(1.6) 295
1.4 The linearized equation of equation (1.17) near non-
trivial steady-states 297
1.4.1 The steady-state equation 297
1.4.2 Linearization of equation (1.17)near (n,N) 298
1.4.3 Exponential solutions of (1.20) 299
1.5 Conclusion 303
2 The Cauchy Problem For An Abstract Linear Delay Differen-
tial Equation 303
2.1 Resolution of the Cauchy problem 304
2.2 Semigroup approach to the problem (CP) 306
2.3 Some results about the range of λI −A 310
3 Formal Duality 311
3.1 The formal adjoint equation 313
∗
3.2 The operator A formal adjointof A 316
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
4.1 Some spectral properties of C -semigroups 321
0
Contents ix
4.2 Decomposition of the state space C([−r,0];E) 324
4.3 A Fredholm alternative principle 326
4.4 CharacterizationofthesubspaceR(λI−A)m forλin
(σ\σ )(A) 326
e
4.5 Characterization of the projection operator onto the
subspace Q 331
Λ
4.6 Conclusion 335
5 AVariationOfConstantsFormulaForAnAbstractFunctional
Differential Equation Of Retarded Type 335
5.1 The nonhomogeneous problem 336
5.2 Semigroup defined in L(E) 337
5.3 The fundamental solution 338
5.4 Thefundamentalsolutionandthenonhomogeneous
problem 341
5.5 Decomposition of the nonhomogeneous problem
in C([−r,0];E) 344
9
TheBasicTheoryofAbstractSemilinearFunctionalDifferential 347
Equations with Non-Dense Domain
K. Ezzinbi and M. Adimy
1 Introduction 347
2 Basic results 350
3 Existence, uniqueness and regularity of solutions 354
4 Thesemigroupandtheintegratedsemigroupintheautonomous
case 372
5 Principle of linearized stability 381
6 Spectral Decomposition 383
7 Existence of bounded solutions 385
8 Existence of periodic or almost periodic solutions 391
9 Applications 393
References 399
Part IV More on Delay Differential Equations and Applications 409
10
Dynamics of Delay Differential Equations 411
H.O. Walther
1 Basic theory and some results for examples 411
1.1 Semiflowsofretardedfunctionaldifferentialequations 411
1.2 Periodic orbits and Poincar´e return maps 416
1.3 Compactness 418
1.4 Global attractors 418
1.5 Linear autonomous equations and spectral
decomposition 419
1.6 Local invariant manifolds for nonlinear RFDEs 423
1.7 Floquet multipliers of periodic orbits 425
1.8 Differential equations with state-dependent delays 435
x DELAY DIFFERENTIAL EQUATIONS
2 Monotone feedback: The structure of invariant sets and
attractors 436
2.1 Negative feedback 437
2.2 Positive feedback 439
3 Chaotic motion 451
4 Stable periodic orbits 456
5 State-dependent delays 468
11
Delay Differential Equations in Single Species Dynamics 477
S. Ruan
1 Introduction 477
2 Hutchinson’s Equation 478
2.1 Stability and Bifurcation 479
2.2 Wright Conjecture 481
2.3 Instantaneous Dominance 483
3 Recruitment Models 484
3.1 Nicholson’s Blowflies Model 484
3.2 Houseflies Model 486
3.3 Recruitment Models 487
4 The Allee Effect 488
5 Food-Limited Models 489
6 Regulation of Haematopoiesis 491
6.1 Mackey-Glass Models 491
6.2 Wazewska-Czyzewska and Lasota Model 493
7 A Vector Disease Model 493
8 Multiple Delays 495
9 Volterra Integrodifferential Equations 496
9.1 Weak Kernel 498
9.2 Strong Kernel 500
9.3 General Kernel 502
9.4 Remarks 504
10 Periodicity 505
10.1 Periodic Delay Models 505
10.2 Integrodifferential Equations 507
11 State-Dependent Delays 511
12 Diffusive Models with Delay 514
12.1 Fisher Equation 514
12.2 Diffusive Equations with Delay 515
12
Well-Posedness, Regularity and Asymptotic Behaviour of Re- 519
tarded Differential Equations by Extrapolation Theory
L. Maniar
1 Introduction 519
2 Preliminaries 521
3 Homogeneous Retarded Differential Equations 525
