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Delay Equations de Odo Diekmann

Delay Equations: Applied Mathematical Sciences, cartea 110

Autor Odo Diekmann, Stephan A. Van Gils, Sjoerd M. V. Lunel, Hans Otto Walther
en Limba Engleză Hardback – 9 mai 1995
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.
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Specificații

ISBN-13: 9780387944166
ISBN-10: 0387944168
Pagini: 552
Ilustrații: XII, 536 p.
Dimensiuni: 160 x 241 x 35 mm
Greutate: 0.94 kg
Ediția:1995
Editura: Springer
Colecția Applied Mathematical Sciences
Seria Applied Mathematical Sciences

Locul publicării:New York, NY, United States

Public țintă

Research

Cuprins

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 Linearage-dependent population growth: elaboration of some of the exercises.- VIII The Hopf bifurcation theorem.- References.- List of symbols.- List of notation.

Notă biografică

Yuri Aleksandrovich Kuznetsov (b. in Alma-Ata, USSR, September 8, 1957) received graduate degree in Theoretical Physics from the Rostov-on-Don State University, USSR, in 1979, and the Ph.D. in Physics and Mathematics from the Institute of Biophysics, USSR Academy of Sciences, Pushchino, Moscow Region, in 1986. From 1979 until 2007, he was affiliated with the Research Computing Center (later: Institute of Mathematical Problems of Biology), USSR /Russian Academy of Sciences, Pushchino, where he held different positions, including that of Senior Researcher. During 1991–1998, he visited as Invited Professor/Researcher IIASA (Laxenburg, Austria), Politecnico di Milano (Italy), CWI (Amsterdam, The Netherlands), and ENS (Paris/Lion, France). Since 1999 he is affiliated with Utrecht University (The Netherlands), being Associate Professor at the Department of Mathematics. In 2011, he was appointed as Professor of Numerical Bifurcation Methods at the University of Twente (Enschede, The Netherlands).

Wolf-Jürgen Beyn, born on April 6, 1949 in Hamburg (Germany), studied Mathematics, Physics, and Computer Science at the Universities of Hamburg and Münster (Germany). He graduated in Mathematics at the University of Münster in 1973 and received his PhD there in 1975. He worked as a Scientific Assistant at the Mathematics Institutes of the Universities of Münster (1976 – 1979) and Konstanz (1979 -1981). In 1981 he received his habilitation in Mathematics at the University of Konstanz, and he has been a lecturer ('Privatdozent') there till 1990. Research stays as visiting professor led him to the University of Calgary (Canada) in 1979, to the California Institute of Technology (Pasadena) in 1986, and to the Universities of New Mexico (Albuquerque) and Missouri (Columbia) during the period 1990-2011. In 1990 he was appointed Associate Professor of Mathematics at Bielefeld University (Germany), where he was promoted to full in 2002. Since then he has been the head of the Numerical Analysis group at the Faculty of Mathematics of Bielefeld University.

Odo Diekmann was born on April 14, 1948, in Diepenveen (the Netherlands). He studied mathematics and physics at the University of Amsterdam and received his PhD from that same university in 1978, with a thesis devoted to mathematical epidemiology of infectious diseases and nonlinear integral equations. He worked at the Centre for Mathematics and Computer Science CWI in Amsterdam from 1974 to 1995. From 1986 to 1995 this was combined with a part-time professorship (joint with J.A.J. Metz) at the Biology Department of Leiden University. In 1995 he became Professor in Applied Mathematics at Utrecht University (The Netherlands).