Spaces of dynamical systems

260 Pages·2019·1.26 MB·English

by  Pilyugin S.Yu

SergeiYu.Pilyugin
SpacesofDynamicalSystems
De Gruyter Studies in
Mathematical Physics
|
Edited by
Michael Efroimsky, Bethesda, Maryland, USA
Leonard Gamberg, Reading, Pennsylvania, USA
Dmitry Gitman,São Paulo, Brazil
Alexander Lazarian, Madison, Wisconsin, USA
BorisSmirnov, Moscow, Russia
Volume 3

Sergei Yu. Pilyugin
Spaces of
Dynamical
Systems
|
2nd edition
PhysicsandAstronomyClassification2010
01.30mm,02.30Hq,05.45.-a,05.45Ac
Author
Prof.Dr.SergeiYu.Pilyugin
St.PetersburgStateUniversity
Universitetskayanab7–9
St.Petersburg199034
Russia
ISBN978-3-11-064446-3
e-ISBN(PDF)978-3-11-065716-6
e-ISBN(EPUB)978-3-11-065399-1
ISSN2194-3532
LibraryofCongressControlNumber:2019938959
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2019WalterdeGruyterGmbH,Berlin/Boston
Coverimage:SciencePhotoLibrary/Heller,Eric
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Contents
Prefacetothefirstedition|IX
Prefacetothesecondedition|XIII
Listofmainsymbols|XV
1 Dynamicalsystems|1
1.1 Maindefinitions|1
1.2 Embeddingofadiscretedynamicalsystemintoaflow|10
1.3 LocalPoincarédiffeomorphism|11
1.4 Time-periodicsystemsofdifferentialequations|13
1.5 Actionofagroup|14
2 Topologiesonspacesofdynamicalsystems|16
2.1 C0-topology|16
2.2 C1-topology|17
2.3 Metricsonthespaceofsystemsofdifferentialequations|18
2.4 Genericproperties|23
2.5 Immersionsandembeddings|24
3 Equivalencerelations|26
3.1 Topologicalconjugacy|26
3.2 Topologicalequivalenceofflows|30
3.3 Nonwanderingset|30
3.4 Localequivalence|36
4 Hyperbolicfixedpoint|37
4.1 Hyperboliclinearmapping|37
4.2 TheGrobman–Hartmantheorem|40
4.3 Neighborhoodofahyperbolicfixedpoint|47
4.4 Thestablemanifoldtheorem|52
4.5 Hyperbolicperiodicpoint|64
5 Hyperbolicrestpointandhyperbolicclosedtrajectory|66
5.1 Hyperbolicrestpoint|66
5.2 Hyperbolicclosedtrajectory|71
VI | Contents
6 Transversality|77
6.1 Transversalityofmappingsandsubmanifolds|77
6.2 Transversalitycondition|79
6.3 Palislemma|81
6.4 Transversalityandhyperbolicityforone-dimensionalmappings|89
7 Hyperbolicsets|90
7.1 Definitionofahyperbolicset|90
7.2 Examplesofhyperbolicsets|92
7.3 Basicpropertiesofhyperbolicsets|95
7.4 Stablemanifoldtheorem|99
7.5 AxiomA|101
7.6 Hyperbolicsetsofflows|109
8 Anosovdiffeomorphisms|116
9 Smale’shorseshoeandchaos|123
9.1 Smale’shorseshoe|123
9.2 Chaoticsets|128
9.3 Homoclinicpoints|129
10 Closinglemma|131
11 C0-genericpropertiesofdynamicalsystems|136
11.1 Hausdorffmetric|136
11.2 Semicontinuousmappings|137
11.3 TolerancestabilityandTakens’theory|138
11.4 Attractorsofdynamicalsystems|142
12 Shadowingofpseudotrajectoriesindynamicalsystems|153
12.1 Definitionsandresults|153
12.2 ProofofTheorem12.1|158
12.3 ProofofTheorem12.2|166
12.4 ProofofTheorem12.3|169
13 Invariantmeasures|174
13.1 Maindefinitions|174
13.2 Constructionofinvariantmeasures|175
13.3 Krylov–Bogolyubovtheorem|179
13.4 Ergodictheorems|180
13.5 PoincarérecurrenceandHamiltoniansystems|189
13.6 Ergodicclosinglemma|195
Contents | VII
A SchemeoftheproofoftheMañétheorem|196
B Lecturesonselectedchaptersofthehistoryofdifferentialequationsand
dynamicalsystems|206
B.1 DifferentialequationsandNewton’sanagram|206
B.2 Developmentofthegeneraltheory|208
B.3 Linearequationsandsystems|212
B.4 Stability|217
B.5 Nonlocalqualitativetheory.Dynamicalsystems|224
B.6 Structuralstability|228
B.7 Dynamicalsystemswithchaoticbehavior|231
Bibliography|236
Index|239