Author | Diekmann, Odo. author |
---|---|
Title | Delay Equations [electronic resource] : Functional-, Complex-, and Nonlinear Analysis / by Odo Diekmann, Sjoerd M. Verduyn Lunel, Stephan A. van Gils, Hanns-Otto Walther |
Imprint | New York, NY : Springer New York, 1995 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4206-2 |
Descript | XII, 536 p. online resource |
0 Introduction and preview -- 0.1 An example of a retarded functional differential equation -- 0.2 Solution operators -- 0.3 Synopsis -- 0.4 A few remarks on history -- I Linear autonomous RFDE -- I.1 Prelude: a motivated introduction to functions of bounded variation -- I.2 Linear autonomous RFDE and renewal equations -- I.3 Solving renewal equations by Laplace transformation -- I.4 Estimates for det ?(z) and related quantities -- I.5 Asymptotic behaviour for t ? ? -- I.6 Comments -- II The shift semigroup -- II.1 Introduction -- II.2 The prototype problem -- II.3 The dual space -- II.4 The adjoint shift semigroup -- II.5 The adjoint generator and the sun subspace -- II.6 The prototype system -- II.7 Comments -- III Linear RFDE as bounded perturbations -- III.1 The basic idea, followed by a digression on weak* integration -- III.2 Bounded perturbations in the sun-reflexive case -- III.3 Perturbations with finite dimensional range -- III.4 Back to RFDE -- III.5 Interpretation of the adjoint semigroup -- III.6 Equivalent description of the dynamics -- III.7 Complexification -- III.8 Remarks about the non-sun-reflexive case -- III.9 Comments -- IV Spectral theory -- IV.1 Introduction -- IV.2 Spectral decomposition for eventually compact semigroups -- IV.3 Delay equations -- IV.4 Characteristic matrices, equivalence and Jordan chains -- IV.5 The semigroup action on spectral subspaces for delay equations -- IV.6 Comments -- V Completeness or small solutions? -- V.l Introduction -- V.2 Exponential type calculus -- V.3 Completeness -- V.4 Small solutions -- V.5 Precise estimates for ??(z)-1? -- V.6 Series expansions -- V.7 Lower bounds and the Newton polygon -- V.8 Noncompleteness, series expansions and examples -- V.9 Arbitrary kernels of bounded variation -- V.10 Comments -- VI Inhomogeneous linear systems -- VI.1 Introduction -- VI.2 Decomposition in the variation-of-constants formula -- VI.3 Forcing with finite dimensional range -- VI.4 RFDE -- VI.5 Comments -- VII Semiflows for nonlinear systems -- VII.1 Introduction -- VII.2 Semiflows -- VII.3 Solutions to abstract integral equations -- VII.4 Smoothness -- VII.5 Linearization at a stationary point -- VII.6 Autonomous RFDE -- VII.7 Comments -- VIII Behaviour near a hyperbolic equilibrium -- VIII.1 Introduction -- VIII.2 Spectral decomposition -- VIII.3 Bounded solutions of the inhomogeneous linear equation -- VIII.4 The unstable manifold -- VIII.5 Invariant wedges and instability -- VIII.6 The stable manifold -- VIII.7 Comments -- IX The center manifold -- IX.1 Introduction -- IX.2 Spectral decomposition -- IX.3 Bounded solutions of the inhomogeneous linear equation -- IX.4 Modification of the nonlinearity -- IX.5 A Lipschitz center manifold -- IX.6 Contractions on embedded Banach spaces -- IX.7 The center manifold is of class Ck -- IX.8 Dynamics on and near the center manifold -- IX.9 Parameter dependence -- IX.10 A double eigenvalue at zero -- IX.11 Comments -- X Hopf bifurcation -- X.l Introduction -- X.2 The Hopf bifurcation theorem -- X.3 The direction of bifurcation -- X.4 Comments -- XI Characteristic equations -- XI.1 Introduction: an impressionistic sketch -- XI.2 The region of stability in a parameter plane -- XI.3 Strips -- XI.4 Case studies -- XI.5 Comments -- XII Time-dependent linear systems -- XII.1 Introduction -- XII.2 Evolutionary systems -- XII.3 Time-dependent linear RFDE -- XII.4 Invariance of X?: a counterexample and a sufficient condition -- XII.5 Perturbations with finite dimensional range -- XII.6 Comments -- XIII Floquet Theory -- XIII.1 Introduction -- XIII.2 Preliminaries on periodicity and a stability result -- XIII.3 Floquet multipliers -- XIII.4 Floquet representation on eigenspaces -- XIII.5 Comments -- XIV Periodic orbits -- XIV.1 Introduction -- XIV.2 The Floquet multipliers of a periodic orbit -- XIV.3 Poincarรฉ maps -- XIV.4 Poincarรฉ maps and Floquet multipliers -- XIV.5 Comments -- XV The prototype equation for delayed negative feedback: periodic solutions -- XV.1 Delayed feedback -- XV.2 Smoothness and oscillation of solutions -- XV.3 Slowly oscillating solutions -- XV.4 The a priori estimate for unstable behaviour -- XV.5 Slowly oscillating solutions which grow away from zero, periodic solutions -- XV.6 Estimates, proof of Theorem 5.5(i) and (iii) -- XV.7 The fixed-point index for retracts in Banach spaces, Whyburnโs lemma -- XV.8 Proof of Theorem 5.5(ii) and (iv) -- XV.9 Comments -- XVI On the global dynamics of nonlinear autonomous differential delay equations -- XVI.1 Negative feedback -- XVI.2 A limiting case -- XVI.3 Chaotic dynamics in case of negative feedback -- XVI.4 Mixed feedback -- XVI.5 Some global results for general autonomous RFDE -- Appendices -- I Bounded variation, measure and integration -- I.1 Functions of bounded variation -- I.2 Abstract integration -- II Introduction to the theory of strongly continuous semigroups of bounded linear operators and their adjoints -- II. 1 Strongly continuous semigroups -- II.2 Interlude: absolute continuity -- II.3 Adjoint semigroups -- II.4 Spectral theory and asymptotic behaviour -- III The operational calculus -- III.1 Vector-valued functions -- III.2 Bounded operators -- III.3 Unbounded operators -- IV Smoothness of the substitution operator -- V Tangent vectors, Banach manifolds and transversality -- V.1 Tangent vectors of subsets of Banach spaces -- V.2 Banach manifolds -- V.3 Submanifolds and transversality -- VI Fixed points of parameterized contractions -- VII Linear age-dependent population growth: elaboration of some of the exercises -- VIII The Hopf bifurcation theorem -- References -- List of symbols -- List of notation