Table Of ContentEMS Tracts in Mathematics 21
EMS Tracts in Mathematics
Editorial Board:
Carlos E. Kenig (The University of Chicago, USA)
Andrew Ranicki (The University of Edinburgh, Great Britain)
Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA)
Vladimir Turaev (Indiana University, Bloomington, USA)
Alexander Varchenko (The University of North Carolina at Chapel Hill, USA)
This series includes advanced texts and monographs covering all fields in pure and applied
mathematics. Tractswill give a reliable introduction and reference to special fields of
current research. The books in the series will in most cases be authored monographs,
although edited volumes may be published if appropriate. They are addressed to graduate
students seeking access to research topics as well as to the experts in the field working
at the frontier of research.
For a complete listing see our homepage at www.ems-ph.org.
,
6 Erich Novak and Henryk Wozniakowski, Tractability of Multivariate Problems.
Volume I: Linear Information
7 Hans Triebel, Function Spaces and Wavelets on Domains
8 Sergio Albeverio et al., The Statistical Mechanics of Quantum Lattice Systems
9 Gebhard Böckle and Richard Pink,Cohomological Theory of Crystals over Function Fields
10 Vladimir Turaev,Homotopy Quantum Field Theory
11 Hans Triebel,Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration
,
12 Erich Novak and Henryk Wozniakowski, Tractability of Multivariate Problems.
Volume II: Standard Information for Functionals
13 Laurent Bessières et al.,Geometrisation of 3-Manifolds
14 Steffen Börm,Efficient Numerical Methods for Non-local Operators. (cid:1)2-Matrix
Compression, Algorithms and Analysis
15 Ronald Brown, Philip J. Higgins and Rafael Sivera,Nonabelian Algebraic Topology.
Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids
16 Marek Janicki and Peter Pflug,Separately Analytical Functions
17 Anders Björn and Jana Björn,Nonlinear Potential Theory on Metric Spaces
,
18 Erich Novak and Henryk Wozniakowski, Tractability of Multivariate Problems.
Volume III: Standard Information for Operators
19 Bogdan Bojarski, Vladimir Gutlyanskii, Olli Martio and Vladimir Ryazanov, Infinitesimal
Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane
20 Hans Triebel, Local Function Spaces, Heat and Navier–Stokes Equations
Kaspar Nipp
Daniel Stoffer
Invariant Manifolds in
Discrete and Continuous
Dynamical Systems
Authors:
Kaspar Nipp Daniel Stoffer
ETH Zürich ETH Zürich
Seminar for Applied Mathematics (SAM) Department of Mathematics
CH-8092 Zürich CH-8092 Zürich
Switzerland Switzerland
E-mail: [email protected] E-mail: [email protected]
2010 Mathematical Subject Classification: 37-02; 37Cxx, 37Dxx, 34Cxx, 34Dxx, 65Lxx, 65P10
Key words: Discrete and continuous dynamical systems, invariant manifolds, foliation, singular
perturbations, geometric numerical integration, differential algebraic equations
ISBN 978-3-03719-124-8
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad-
casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use
permission of the copyright owner must be obtained.
© European Mathematical Society 2013
Contact address:
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Email: [email protected]
Homepage: www.ems-ph.org
Typeset using the authors’ TEX files: I. Zimmermann, Freiburg
Printed in Germany
9 8 7 6 5 4 3 2 1
Preface
Inthisbookdynamicalsystemsareinvestigatedfromageometricviewpoint. Astrong
geometricpropertyofadynamicalsystemisadmittinganinvariantmanifold. Inthis
case essential dynamics of the system takes place on some lower dimensional sur-
face. Invariantmanifoldsareanimportanttoolinabroadrangeofapplicationssuch
as mechanical systems, chemical reaction dynamics, fluid mechanics, electronic cir-
cuit theory and singular perturbation theory. The theory of invariant manifolds for
dynamicalsystemsiswellestablished. ItgoesbacktotheworkofHadamard[47]and
Perron[105]anditwasfurtherdevelopedbymanyauthors. WementionFenichel[38],
[39],[40],Hirsch,Pugh,Shub[55],Kelley[63],Carr[23],Wiggins[129],Chaperon
[24], [25]. The aim of this book is to present rigorous results on invariant manifolds
indynamicalsystemsandtogiveexamplesofpossibleapplications. Thebookistar-
geted at researchers in the field of dynamical systems interested in precise theorems
easytoapply. PartIIImightalsoserveasanunderlyingtextforastudentseminarin
mathematics.
Ourapproachtoinvariantmanifoldsfordiscretedynamicalsystemsisbasedonthe
so-calledgraphtransform,alreadyusedbyHadamard[47].Assumethatthedynamical
systemisgivenbyamapPW X(cid:2)Y !X(cid:2)Y whereX,Y arenonemptyopensetsin
someBanachspaces. WeconsidermanifoldsdescribedasthegraphM ofafunction
(cid:2)W X ! Y. Under certain conditions the image of M under the map P is again a
graphofafunction(cid:2)NW X !Y. ThisinducesanoperatorF inthespaceoffunctions
considered. AninvariantmanifoldisobtainedasafixedpointoftheoperatorF. The
existenceofafixedpointisestablishedbythecontractionprinciple.
Wealsoconsidercontinuousdynamicalsystemsgivenbyanordinarydifferential
equation (ODE). The time-T map of an ODE is a discrete dynamical system. The
graphtransformapproachformapscarriesovertoODEs,ifappliedtothetime-T map.
Wegiveconditionsonthevectorfieldimplyingtheexistenceofaninvariantmanifold
for the ODE. Invariant manifold results for ODEs can also be derived by different
approacheswithoutusingthegraphtransform. WementionPerron[105],Kelley[63],
Knobloch,Kappel[72],Knobloch[69],Yi[130].
In the existing literature on invariant manifolds there is a strong tendency to for-
mulatetheresultsinarathergeneralsetting. Weaimatinvariantmanifoldresultsina
simplesetting,easytoapplyandprovidingquantitativeestimates. AsinKirchgraber,
Lasagni,Nipp,Stoffer[66]weformulateconditionseasytoverifyandleadingtosharp
resultsifthecoordinatesarechoseninanappropriateway. Inthediscretecasewegive
conditionsontheLipschitzconstantsofthemapandinthecontinuouscaseconditions
onthederivativesofthevectorfield.
Thebookisorganizedasfollows. InPartIdiscretedynamicalsystemsinBanach
spaces are considered. We derive results on the existence of attractive and repulsive
invariantmanifoldsthatmaybedescribedasthegraphofsomefunction. Wealsotreat
vi Preface
manifoldsdescribedinseveralcharts. Inaddition,westateresultsonthesmoothness
oftheinvariantmanifold,onperturbationsofthemanifoldandonthefoliationofthe
adjacent space. In Part II we establish analogous results for continuous dynamical
systemsinfinitedimensions. InPartIIIweapplytheresultsofthefirsttwoparts. The
emphasisisonapplicationstonumericalanalysisandtosingularlyperturbedsystems
of ODEs. In an appendix the hypotheses and conditions used in the theorems are
arrangedtohelptonavigateinthebulkofassumptionsmade.
We want to thank several people who supported us in this book project. Special
thanks go to Urs Kirchgraber who many years back introduced us to the topic of
invariant manifolds for maps and to the graph transform approach. We also thank
PeterSzmolyanwhoinitiatedourresearchongeometricsingularperturbationtheory
formapsaspresentedinChapter13. Thefinishofthematerialofthisbookaswellas
partofthefinaleditingwasdoneonasabbaticalleaveattheTUBerlininthesummer
of 2011. We thank the ETH for this support andVolker Mehrmann for hosting us at
his institute and for providing a good working atmosphere. We also thank Marianne
PfisterfortransformingalargepartofourhandwritingintoLATEXcodeandOlivier
Barbey for drawing part of the figures. The EMS publishing house and in particular
ManfredKarbeandIreneZimmermannhaveefficientlyhandledourmanuscript,many
thanks.
Zürich,June2012 KasparNipp
DanielStoffer
Contents
Preface v
I DiscreteDynamicalSystems–Maps 1
1 Existence 5
1.1 Repulsivepositivelyinvariantmanifolds . . . . . . . . . . . . . . . 5
1.2 Attractivenegativelyinvariantmanifolds . . . . . . . . . . . . . . . 13
1.3 Hyperbolicinvariantmanifolds . . . . . . . . . . . . . . . . . . . . 20
1.4 Manifoldsdefinedbyseveralcharts . . . . . . . . . . . . . . . . . 24
1.4.1 Ageneralexistenceresult . . . . . . . . . . . . . . . . . . . 24
1.4.2 ToolsforChapter13 . . . . . . . . . . . . . . . . . . . . . 29
2 Perturbationandapproximation 33
2.1 Attractivenegativelyinvariantmanifolds . . . . . . . . . . . . . . . 33
2.2 Repulsivepositivelyinvariantmanifolds . . . . . . . . . . . . . . . 35
3 Smoothness 37
3.1 Repulsivepositivelyinvariantmanifolds . . . . . . . . . . . . . . . 37
3.1.1 Thefirstderivative . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Thehigherderivatives . . . . . . . . . . . . . . . . . . . . 41
3.1.3 Thedifferentiabilitywithrespecttoparameters . . . . . . . 44
3.2 Attractivenegativelyinvariantmanifolds . . . . . . . . . . . . . . . 45
4 Foliation 46
4.1 Thestablefoliation . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Theunstablefoliation . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Thehyperboliccase . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Smoothnessofthefoliationwithrespecttothebasepoint 59
5.1 Thestablefoliation . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Theunstablefoliation . . . . . . . . . . . . . . . . . . . . . . . . . 68
II ContinuousDynamicalSystems–ODEs 69
6 Ageneralresultforthetime-T map 71
7 Invariantmanifoldresults 74
7.1 Auxiliaryresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2 Attractivenegativelyinvariantmanifolds . . . . . . . . . . . . . . . 81
7.3 Repulsivepositivelyinvariantmanifolds . . . . . . . . . . . . . . . 88
viii Contents
III Applications 93
8 Fixedpointsandequilibria 95
8.1 Thelocalstableandunstablemanifoldofahyperbolicfixedpoint . 95
8.2 Thestronglystablemanifoldofanequilibrium . . . . . . . . . . . 99
9 Theone-stepmethodassociatedtoalinearmultistepmethod 101
9.1 Basicfactsonlinearmultistepmethods . . . . . . . . . . . . . . . 102
9.2 Theassociatedone-stepmethod . . . . . . . . . . . . . . . . . . . 103
9.3 Theglobalerroroflinearmultistepmethods . . . . . . . . . . . . . 108
10 InvariantmanifoldsforsingularlyperturbedODEs 110
10.1 Attractivemanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 110
10.2 Hyperbolicmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 114
11 Runge–KuttamethodsappliedtosingularlyperturbedODEs 121
11.1 Nonstiffmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
11.2 Stiffmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
11.2.1 Theinvariantmanifold . . . . . . . . . . . . . . . . . . . . 126
11.2.2 Theglobalerror . . . . . . . . . . . . . . . . . . . . . . . . 133
12 Invariantcurvesofperturbedharmonicoscillators 137
12.1 ThevanderPolequation . . . . . . . . . . . . . . . . . . . . . . . 137
12.1.1 Themethodofaveragingforperturbedharmonicoscillators 139
12.1.2 Theinvariantmanifoldforperturbedharmonicoscillators . . 140
12.1.3 ApplicationtothevanderPolequation . . . . . . . . . . . 142
12.2 ThesymplecticEulermethod . . . . . . . . . . . . . . . . . . . . . 143
12.2.1 Themethodofaveragingforthemap . . . . . . . . . . . . 146
12.2.2 Theinvariantmanifoldofthemap . . . . . . . . . . . . . . 148
12.2.3 ApplicationtothevanderPolequation . . . . . . . . . . . 149
12.3 TheEulermethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
13 Blow-upinsingularperturbations 156
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
13.2 Themainresult . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
13.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
13.3.1 Theblow-up. . . . . . . . . . . . . . . . . . . . . . . . . . 159
13.3.2 Thereferencemanifold . . . . . . . . . . . . . . . . . . . . 160
13.4 Proofofthemainresult . . . . . . . . . . . . . . . . . . . . . . . . 161
13.4.1 Thechartˆ1 . . . . . . . . . . . . . . . . . . . . . . . . . 162
13.4.2 Thechartˆ2 . . . . . . . . . . . . . . . . . . . . . . . . . 165
13.4.3 Thechartˆ3 . . . . . . . . . . . . . . . . . . . . . . . . . 170
13.4.4 Thechartˆ4 . . . . . . . . . . . . . . . . . . . . . . . . . 174
Contents ix
13.4.5 Thechartˆ5 . . . . . . . . . . . . . . . . . . . . . . . . . 178
13.4.6 Thechartˆ6 . . . . . . . . . . . . . . . . . . . . . . . . . 182
14 ApplicationofRunge–Kuttamethodstodifferential-algbraicequations 187
IV Appendices 197
A Hypothesesandconditionsformaps 199
A.1 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
A.2 Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
B HypothesesandconditionsforODEs 204
B.1 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
B.2 Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Bibliography 207
Index 215
