Delay Equations : Functional-, Complex-, and Nonlinear Analysis / by Odo Diekmann, Sjoerd M. Verduyn Lunel, Stephan A. van Gils, Hanns-Otto Walther
(Applied Mathematical Sciences ; 110)
| データ種別 | 電子ブック |
|---|---|
| 出版者 | New York, NY : Springer New York |
| 出版年 | 1995 |
| 本文言語 | 英語 |
| 大きさ | XII, 536 p : online resource |
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| 内容注記 | 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 Poincaré maps XIV.4 Poincaré 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, Whyburn’s 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 |
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| 一般注記 | The aim of this book is to provide an introduction to the mathematical theory of infinite dimensional dynamical systems by focusing on a relatively simple, yet rich, class of examples, that is, those described by delay differential equations. It is a textbook giving detailed proofs and providing many exercises, which is intended both for self-study and for courses at a graduate level. The book would also be suitable as a reference for basic results. As the subtitle indicates, the book is about concepts, ideas, results and methods from linear functional analysis, complex function theory, the qualitative theory of dynamical systems and nonlinear analysis. After studying this book, the reader should have a working knowledge of applied functional analysis and dynamical systems |
| 著者標目 | *Diekmann, Odo author Verduyn Lunel, Sjoerd M. author Gils, Stephan A. van author Walther, Hanns-Otto author SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Mathematical analysis LCSH:Analysis (Mathematics) LCSH:Dynamics LCSH:Ergodic theory LCSH:Mathematics -- Study and teaching 全ての件名で検索 FREE:Mathematics FREE:Analysis FREE:Mathematics Education FREE:Dynamical Systems and Ergodic Theory |
| 分 類 | DC23:515 |
| 巻冊次 | ISBN:9781461242062 
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| ISBN | 9781461242062 |
| URL | http://dx.doi.org/10.1007/978-1-4612-4206-2 |
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