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** Delay Differential Equations and Applications ** edited by ** O. Arino **, University of Pau, France and
** M.L. Hbid **, University Cadi Ayyad, Marrakech, Morocco and
** E. Ait Dads**, University Cadi Ayyad, Marrakech, Morocco.

** Delay Differential Equations and Applications ** edited by
** O. Arino **,
** M.L. Hbid **and
** E. Ait Dads**
cover the following topics of Differential Equations.

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

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

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

References

1 Introduction

2 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

3 Variation Of Constant Formula Using Integrated Semigroups Theory

3.1 Notations and basic results

3.2 The variation of constant formula

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

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

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

4 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

5 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

1 Introduction 285

1.1 A model of fish population dynamics with spatial diffusion (11)

1.2 An abstract differential equation arising from cell population dynamics

1.3 From integro-difference to abstract delay differential equations (8)

1.3.1 The linear equation 292

1.3.2 Delay differential equation formulation of system (1.5)-(1.6)

1.4 The linearized equation of equation (1.17) near nontrivial steady-states

1.4.1 The steady-state equation

1.4.2 Linearization of equation (1.17) near (n, N)

1.4.3 Exponential solutions of (1.20)

1.5 Conclusion

2 The Cauchy Problem For An Abstract Linear Delay Differential Equation

2.1 Resolution of the Cauchy problem

2.2 Semigroup approach to the problem (CP)

2.3 Some results about the range of ?I - A

3 Formal Duality

3.1 The formal adjoint equation

3.2 The operator A* formal adjoint of A

3.3 Application to the model of cell population dynamics

3.4 Conclusion

4 Linear Theory Of Abstract Functional Differential Equations Of Retarded Type

4.1 Some spectral properties of C0-semigroups

4.3 A Fredholm alternative principle

4.4 Characterization of the subspace R (?I -A)m for ? in (s\se)(A)

4.5 Characterization of the projection operator onto the subspace Q?

4.6 Conclusion

5 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)

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

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

2 The structure of invariant sets and attractors

2.1 Negative feedback

2.2 Positive feedback

3 Chaotic motion

4 Stable periodic orbits

5 State-dependent delays

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

1 Introduction

2 Preliminaries

3 Homogeneous Retarded Differential Equations

Index

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

1 Program listings

1.1 MatLab code

1.2 XPPAUT code

2 Delay differential equations packages

2.1 Numerical integration

2.2 Bifurcation analysis

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