An Introduction to Infinite Dimensional Dynamical Systems — Geometric Theory

203 Pages·1984·4.096 MB·English

by  Jack K. Hale, Luis T. Magalhães, Waldyr M. Oliva (auth.)

Applied
Mathematical
Sciences
EDITORS
FIttzJohn J.E. Marsden Lawrence Sirovich
Courant Institute of Department of Division of
Mathematical ScIences Mathematics Applied Mathematics
New York University University of Califomia Brown University
NewYork,NY10012 Berkeley, CA 94720 Providence, RI 02912
ADVISORS
H. Cabannes University of Paris-VI J. Keller Stanford University
M. Ghl New York University J.P. LaSalle Brown University
J.K. Hale Brown University G.B. Whitham Califomia Inst. of Technology
EDITORIAL STATEMENT
The mathematization of all sciences, the fading of traditional scientific bounda
ries, the impact of computer technology, the growing importance of mathematical
computer modelling and the necessity of scientific planning all create the need both
in education and research for books that are introductory to and abreast of these
developments.
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tional format and in hard cover.
MANUSCRIPTS
The Editors welcome ali inquiries regarding the submission of manuscripts for
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SPRINGER-VERLAG NEW -YORK INC., 175 Fifth Avenue, New York, N. Y. 10010
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I
Applied Mathematical Sciences Volume 47
Applied Mathematical Sciences
1. John: Partial Differential Equations, 4th ed. (cloth)
2. Sirovich: Techniques of Asymptotic Analysis.
3. Hale: Theory of Functional Differential Equations, 2nd ed. (cloth)
4. Percus: Combinatorial Methods.
5. von Mises/Friedrichs: Fluid Dynamics.
6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics.
7. Pipkin: Lectures on Viscoelasticity Theory.
8. Giacaglia: Perturbation Methods in Non-Linear Systems.
9. Friedrichs: Spectral Theory of Operators in Hilbert Space.
10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations.
11. Wolovich: Linear Multivariable Systems.
12. Berkovitz: Optimal Control Theory.
13. Bluman/Cole: Similarity Methods for Differential Equations.
14. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost
Periodic Solutions.
15. Braun: Differential Equations and Their Applications, 3rd ed. (cloth)
16. Lefschetz: Applications of Algebraic Topology.
17. Collatz/Wetterling: Optimization Problems.
18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I.
19. Marsden/McCracken: The Hopf Bifurcation and its Applications.
20. Driver: Ordinary and Delay Differential Equations.
21. Courant/Friedrichs: Supersonic Flow and Shock Waves. (cloth)
22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method.
23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory.
24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II.
25. Davies: Integral Transforms and Their Applications.
26. Kushner/Clark: Stochastic Approximation Methods for Constrained and
Unconstrained Systems.
27. de Boor: A Practical Guide to Splines.
28. Keilson: Markov Chain Models-Rarity and Exponentiality.
29. de Veuoeke: A Course in Elasticity.
30. Sniatycki: Geometric Quantization and Quantum Mechanics.
31. Reid: Sturmian Theory for Ordinary Differential Equations.
32. Meis/Markowitz: Numerical Solution of Partial Differential Equations.
33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III.
34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics. (cloth)
35. Carr: Applications of Centre Manifold Theory.
(continued after Index)
Jack K. Hale
Luis T.
Magalhăes
Waldyr M. Oliva
An Introduction to Infinite
Dimensional Dynamical
Systems-
Geometric
Theory
With an Append ix by Krzysztof P. Rybakowski
With 17 IlIustrations
Springer-Verlag Berlin Heidelberg GmbH
Jack K. Hale Luis T. Magalhăes
Division of Applied Mathematics Universidade Tecnica de Lisboa
Brown University Lisbon
Providence, R.1. 02912 Portugal
U.S.A.
Krzysztof P. Rybakowski
Waldyr M. Oliva Technische Universitat Berlin
Departmento de Matematica Aplicada Berlin
Instituto de Matematica e Estatistica Federal Republic of Germany
Universidade de Săo Paulo
Săo Paulo
Brasil
AMS Subject Classifications: 34C35, 54H20
Library of Congress Cataloging in Publication Data
Hale, Jack K.
An introduction to infinite dimensional dynamical
systems -geometric theory.
(Applied mathematical sciences; v. 47)
Bibliography: p.
Includes index.
1. Differentiable dynamical systems. 1. Magalhăes,
Luis T. II. Oliva, Waldyr M. III. Title. IV. Series:
Applied mathematica1 sciences (Springer-Verlag New York
Inc.) ; v. 47.
Qa1.A647 voI. 47 [QA614.8] 510s [514/ .7] 83-20043
© 1984 by Springer-Verlag Berlin Heidelberg
Origina1ly published by Springer-Verlag Berlin Heidelberg New York Tokyo in 1984
Ali rights reserved. No part ofthis book may be translated or reproduced in any form
without written permis sion from Springer-Verlag Berlin Heidelberg GmbH
98765432
ISBN 978-0-387-90931-8 ISBN 978-1-4757-4493-4 (eBook)
DOI 10.1007/978-1-4757-4493-4
Preface
The motivation for writing these notes came from a series of lectures
of the third author on retarded functional differential equations at the
Lefschetz Center for Dynamical Systems of the Division of Applied Mathematics
at Brown University during the spring of 1982. Partial financial support
was obtained from the Air Force Office of Scientific Research, AF-AFOSR 81-
0198, National Science Foundation, MCS 79-05774-05, U. S. Army Research
Office, DAAG-29-79-C-016l, Instituto Nacional de Investiga~ao Cientifica,
Portugal, Conselho Nacional de Desenvolvimento Cientifico e Tecnologico
(CNPq), Proc. No. 40.3278/81, Fapesp and Reitoria da Universidade de Sao
Paulo, Brasil.
The authors appreciate the efforts of Dorothy Libutti, Katherine
MacDougall, and Nancy Gancz for the preparation of the manuscript.
Contents
1. Introduction 1
2. Retarded Functional Differential Equations on Manifolds 7
3. Examples of Retarded Functional Differential Equations on
Manifolds 13
4. Generic Properties. The Theorem of Kupka-Smale 24
5. Invariant Sets, Limit Sets and the Attractor 43
6. The Dimension of the Attractor 56
7. Attractor Sets as Cl-Manifolds 69
8. Stability Relative to A(F) and Bifurcation 85
9. Cornpactification at Infinity 100
10. Stability of Morse-Smale Maps III
11. Bibliographical Notes 140
References 143
Appendix - An Introduction to Homotopy Index
Theory in Noncornpact Spaces 147
References for Appendix 191
Subject Index 193
1. Introduction
Many applications involve dynamical systems in non-locally compact
infinite dimensional spaces; for example, dynamical systems generated by partial
differential equations and delay differential or functional differential equa
tions. Because of the complexities involved in doing detailed analysis in
infinite dimensions, these systems often are approximated by finite dimen
sional dynamical systems generated by ordinary differential equations in Rn
or on a n-dimensional manifold M. The global theory of such dynamical sys
tems is then used to better understand the complete dynamics of the system
and the way that the system behaves as physical parameters are varied.
Although some efforts are being made to extend the finite dimensional
ideas to infinite dimensions, the global theory is still in its infancy.
One reason the development has been so slow follows from the infinite di
mensionality of the problems and the complexities that result from this fact.
Probably a more important reason is a consequence of the fact that the per
sons who work in abstract finite dimensional dynamical systems are unaware
of some of the "nice" systems that exist in infinite dimensions, systems
whose basic structure may be amenable to a mathematical theory approaching
the completeness that is known for finite dimensions and, at the same time,
require new ideas of a fundamental character. On the other side, persons
dealing with specific infinite dimensional problems in the applications often
are not aware of the fact that detailed knowledge of the ideas in finite
dimensional problems can be adapted to their problems.
The purpose of these notes is to outline an approach to the develop
ment of a theory of dynamical systems in infinite dimensions which is
-2-
analogous to the theory of finite dimensions. The first problem is to find
a class for which there is some hope of classification and yet general
enough to include some interesting applications. Throughout the notes, the
discussion centers around retarded functional differential equations al-
though the techniques and several of the results apply to more general situa-
tions; in particular, to neutral functional differential equations, parabolic
partial differential equations and some other types of partial differential
equations.
In the introduction, we give an abstract formulation of a class of
dynamical systems which occur frequently in the applications and state some
of the basic properties and problems that should be studied.
Let X, Y, Z be Banach spaces (sometimes Banach manifolds) and let
~r = Cr(y,Z), r ~ I, be the set of functions from Y to Z which are
bounded and uniformly continuous together with their derivatives up through
order r. We impose the usual topology on~r. (In applications, other
topologies may be needed; for example, the Whitney topology.) For each
f~, let Tf(t): X + X, t ~ 0, be a strongly continuous semigroup of
transformations on X. For each x € X, we suppose Tf(t)x is defined for
t > 0 and is Cr in x.
We say a point xo € X has a backward extension if there is a
q>: (-"',0] + X such that q>(0) = xo and Tf(t)q>(t) = q>(t+t) for 0 < t ~ -t.
t < o. If there is a backward extension q> through xO' we define Tf(t)xO
q>Ct), t < O. A set Me X is invariant if, for each x € M, Tf(t)x is
defined and belongs to M for t € (-"',"'). The orbit Y+Cx) through x
is defined as
-3-
Let
A(f) = {x € X: Tf(t)x is defined and bounded for t € (-~,~)}.
The set A(f) contains much of the interesting information about the semi
group Tf(t). In fact, it is very easy to verify the following result.
Proposition 1.1. If A(f) is compact, then A(f) is maximal, compact,
invariant. If, in addition, all orbits have compact closure, then A(f) is
a global attractor. Finally, if Tf(t) is one-to-one on A(f), then Tf(t)
is a continuous group on A(f).
The first difficulty in infinite dimensional systems is to decide how
to compare two semigroups Tf(t), Tg(t). It seems to be almost impossible
to make a comparison of any system on all or even an arbitrary bounded set
of X. If A(f) is compact, Proposition 1.1 indicates that all essential
information is contained in A(f). Thus, we define equivalence relatively
to A(f).
Definition 1.2. We say f is equivalent to g, f ~ g, if there is a
homeomorphism h: A(f) + A(g) which preserves orbits and the sense of di
rection in time. We say f is stable relative to A(f) or A-stable if
there is a neighborhood V of f in ~r such that g € V implies g ~ f.
We say f is a bifurcation point if f is not A-stable.
The basic problem is to discuss detailed properties of the set A(f)
and to determine how A(f) and the structure of the flow on A(f) change
with f.
If A(f) is not compact, very little is known at this time. It be
comes important therefore to isolate a class of semigroups for which A(f)