Delay differential equations and applications

Contents......Page 6
List of Figures......Page 13
Preface......Page 16
Contributing Authors......Page 19
Introduction......Page 21
1 History Of Delay Equations......Page 25
1 Stability of equilibria and Lyapunov functions......Page 27
2 Invariant Sets, Omega-limits and Lyapunov functionals......Page 31
3 Delays may cause instability......Page 34
4 Linear autonomous equations and perturbations......Page 36
5 Neutral Functional Differential Equations......Page 40
6 Periodically forced systems and discrete dynamical systems......Page 44
7 Dissipation, maximal compact invariant sets and attractors......Page 45
8 Stationary points of dissipative flows......Page 48
Part I General Results and Linear Theory of Delay Equations in Finite Dimensional Spaces......Page 53
1 Introduction......Page 54
2 A general initial value problem......Page 56
1 Basic Theory......Page 64
2 Eigenspaces......Page 94
3 Small Solutions and Completeness......Page 127
4 Degenerate delay equations......Page 133
Appendix: A......Page 150
Appendix: B......Page 152
Appendix: C......Page 154
Appendix: D......Page 155
References......Page 160
Part II Hopf Bifurcation, Centre manifolds and Normal Forms for Delay Differential Equations......Page 163
1 Introduction......Page 164
2 Variation Of Constant Formula Using Sun-Star Machinery......Page 166
3 Variation Of Constant Formula Using Integrated Semigroups Theory......Page 170
1 Introduction......Page 181
2 The Lyapunov Direct Method And Hopf Bifurcation: The Case Of Ode......Page 186
3 The Center Manifold Reduction Of DDE......Page 188
4 Cases Where The Approximation Of Center Manifold Is Needed......Page 202
1 Introduction......Page 212
2 Notations and background......Page 214
3 Computational scheme of a local center manifold......Page 218
4 Computational scheme of Normal Forms......Page 232
1 Introduction......Page 246
2 Normal Forms for FDEs in Finite Dimensional Spaces......Page 250
3 Normal forms and Bifurcation Problems......Page 262
4 Normal Forms for FDEs in Hilbert Spaces......Page 272
5 Normal Forms for FDEs in General Banach Spaces......Page 281
References......Page 294
Part III Functional Differential Equations in Infinite Dimensional Spaces......Page 302
1 Introduction......Page 303
2 The Cauchy Problem For An Abstract Linear Delay Differential Equation......Page 321
3 Formal Duality......Page 329
4 Linear Theory Of Abstract Functional Differential Equations Of Retarded Type......Page 338
5 A Variation Of Constants Formula For An Abstract Functional Differential Equation Of Retarded Type......Page 353
1 Introduction......Page 365
2 Basic results......Page 368
3 Existence, uniqueness and regularity of solutions......Page 372
4 The semigroup and the integrated semigroup in the autonomous case......Page 390
5 Principle of linearized stability......Page 399
6 Spectral Decomposition......Page 401
7 Existence of bounded solutions......Page 403
8 Existence of periodic or almost periodic solutions......Page 409
9 Applications......Page 411
References......Page 416
Part IV More on Delay Differential Equations and Applications......Page 425
1 Basic theory and some results for examples......Page 426
2 Monotone feedback: The structure of invariant sets and attractors......Page 451
3 Chaotic motion......Page 466
4 Stable periodic orbits......Page 471
5 State-dependent delays......Page 483
1 Introduction......Page 492
2 Hutchinson's Equation......Page 493
3 Recruitment Models......Page 499
4 The Allee Effect......Page 503
5 Food-Limited Models......Page 504
6 Regulation of Haematopoiesis......Page 506
7 A Vector Disease Model......Page 508
8 Multiple Delays......Page 510
9 Volterra Integrodifferential Equations......Page 511
10 Periodicity......Page 520
11 State-Dependent Delays......Page 526
12 Diffusive Models with Delay......Page 529
1 Introduction......Page 533
2 Preliminaries......Page 535
3 Homogeneous Retarded Differential Equations......Page 539
1 Introduction......Page 553
2 Origin of time delays in epidemic models......Page 554
3 A model that includes a vaccinated state......Page 558
4 Reduction of the system by using specific P(t) functions......Page 562
5 Numerical considerations......Page 564
6 A few words of warning......Page 566
Appendix......Page 569
References......Page 573
E......Page 593
S......Page 594
V......Page 595