Dimension theory for ordinary differential equations:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Wiesbaden
Teubner
2005
|
| Edition: | 1. ed. |
| Series: | Teubner-Texte zur Mathematik
141 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 441 S. |
| ISBN: | 3519004372 |
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MARC
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Record in the Search Index
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|---|---|
| adam_text | VLADIMIR A. BOICHENKO, GENNADIJ A. LEONOV, VOLKER REITMANN DIMENSION
THEORY FOR ORDINARY DIFFERENTIAL EQUATIONS TEUBNER CONTENTS SINGULAR
VALUES, EXTERIOR CALCULUS AND LOZINSKII-NORMS 15 1 SINGULAR VALUES AND
COVERING OF ELLIPSOIDS 15 1.1 INTRODUCTION 15 1.2 DEFINITION OF SINGULAR
VALUES 17 1.3 LEMMAS ON COVERING OF ELLIPSOIDS 19 2 SINGULAR VALUE
INEQUALITIES 21 2.1 THE FISCHER-COURANT THEOREM 21 2.2 THE BINET-CAUCHY
THEOREM 24 2.3 THE INEQUALITIES OF HORN, WEYL AND FAN 28 3 COMPOUND
MATRICES 30 3.1 MULTIPLICATIVE COMPOUND MATRICES 30 3.2 ADDITIVE
COMPOUND MATRICES 35 3.3 APPLICATIONS TO STABILITY THEORY 38 4
LOGARITHMIC MATRIX NORMS 40 4.1 LOZINSKII S THEOREM 40 4.2
GENERALIZATION OF THE LIOUVILLE EQUATION 45 5 THE YAKUBOVICH-KALMAN
FREQUENCY THEOREM 49 5.1 THE FREQUENCY THEOREM FOR ODE S 49 5.2 THE
FREQUENCY THEOREM FOR DISCRETE-TIME SYSTEMS 52 6 FREQUENCY-DOMAIN
ESTIMATION OF SINGULAR VALUES 54 6.1 LINEAR DIFFERENTIAL EQUATIONS 54
6.2 LINEAR DIFFERENCE EQUATIONS 58 7 EXTERIOR CALCULUS IN LINEAR SPACES
62 7.1 MULTIPLICATIVE AND ADDITIVE COMPOUNDS OF OPERATORS ... 62 7.2
SINGULAR VALUES OF AN OPERATOR ACTING BETWEEN EUCLIDEAN SPACES 72 7.3
LEMMAS ON COVERING OF ELLIPSOIDS IN AN EUCLIDEAN SPACE . 75 7.4 SINGULAR
VALUE INEQUALITIES FOR OPERATORS 76 10 CONTENTS 11 ATTRACTORS, STABILITY
AND LYAPUNOV FUNCTIONS 79 1 DYNAMICAL SYSTEMS, LIMIT SETS AND ATTRACTORS
79 1.1 DYNAMICAL SYSTEMS IN METRIC SPACES 79 1.2 MINIMAL GLOBAL
ATTRACTORS 85 1.3 TIME-INVARIANT CONTROL SYSTEMS 89 2 DISSIPATIVITY 91
2.1 DISSIPATIVITY IN THE SENSE OF LEVINSON 91 2.2 DISSIPATIVITY AND
COMPLETENESS OF THE LORENZ SYSTEM ... 92 2.3 LYAPUNOV-TYPE RESULTS FOR
DISSIPATIVITY 96 2.4 CONVERGENCE IN SYSTEMS WITH SEVERAL EQUILIBRIUM
STATES . 100 3 STABILITY OF MOTION 106 3.1 LYAPUNOV STABILITY 107 3.2
ORBITAL STABILITY 115 3.3 ZHUKOVSKII STABILITY 119 4 EXISTENCE OF A
HOMOCLINIC ORBIT IN THE LORENZ SYSTEM 124 4.1 INTRODUCTION 124 4.2
ESTIMATES FOR THE SHAPE OF GLOBAL ATTRACTORS 125 4.3 THE EXISTENCE OF
HOMOCLINIC ORBITS 127 5 THE GENERALIZED LORENZ SYSTEM 132 5.1 DEFINITION
OF THE SYSTEM 132 5.2 EQUILIBRIUM STATES 133 5.3 GLOBAL ASYMPTOTIC
STABILITY 134 5.4 DISSIPATIVITY 136 6 ORBITAL STABILITY FOR FLOWS ON
MANIFOLDS 138 6.1 INTRODUCTION 138 6.2 DYNAMICAL SYSTEMS WITH A LOCAL
CONTRACTION PROPERTY . . 139 6.3 THE ANDRONOV-VITT THEOREM 143 6.4
VARIOUS TYPES OF VARIATIONAL EQUATIONS 144 6.5 ASYMPTOTIC ORBITAL
STABILITY CONDITIONS 147 6.6 ESTIMATING THE SINGULAR VALUES AND ORBITAL
STABILITY . . . 162 6.7 FREQUENCY-DOMAIN CONDITIONS FOR ORBITAL
STABILITY IN FEEDBACK CONTROL EQUATIONS ON THE CYLINDER 169 III
INTRODUCTION TO DIMENSION THEORY 175 1 TOPOLOGICAL DIMENSION 175 1.1 THE
INDUCTIVE TOPOLOGICAL DIMENSION 176 1.2 THE COVERING DIMENSION 182 2
HAUSDORFF AND FRACTAL DIMENSIONS 186 2.1 THE HAUSDORFF MEASURE AND THE
HAUSDORFF DIMENSION . .186 2.2 FRACTAL DIMENSION AND LOWER BOX DIMENSION
196 CONTENTS 11 2.3 SELF-SIMILAR SETS 202 2.4 DIMENSION OF CARTESIAN
PRODUCTS 205 3 TOPOLOGICAL ENTROPY 207 3.1 THE BOWEN-DINABURG DEFINITION
208 3.2 THE CHARACTERIZATION BY OPEN COVERS 211 3.3 SOME PROPERTIES OF
THE TOPOLOGICAL ENTROPY 214 4 DIMENSION-LIKE CHARACTERISTICS 219 4.1
CARATHEODORY MEASURE, DIMENSION AND CAPACITY 219 4.2 PROPERTIES OF THE
CARATHEODORY DIMENSION AND CARATHEODORY CAPACITY 223 IV DIMENSION AND
LYAPUNOV FUNCTIONS 229 1 ESTIMATION OF THE TOPOLOGICAL DIMENSION 229 1.1
HILMY S THEOREM 229 1.2 MINIMAL SETS FOR ALMOST PERIODIC FLOWS 230 1.3
THE FREQUENCY SPECTRUM OF ALMOST PERIODIC SOLUTIONS . . 236 1.4
FREQUENCY-DOMAIN CONDITIONS FOR UPPER TOPOLOGICAL DIMENSION ESTIMATES OF
ORBIT CLOSURES 242 2 UPPER ESTIMATES FOR THE HAUSDORFF DIMENSION 244 2.1
THE LIMIT THEOREM FOR HAUSDORFF MEASURES 244 2.2 COROLLARIES OF THE
LIMIT THEOREM FOR HAUSDORFF MEASURES . 250 2.3 APPLICATION OF THE LIMIT
THEOREM TO THE HENON MAP . . . 255 3 THE APPLICATION OF THE LIMIT
THEOREM TO ODE S 260 3.1 AN AUXILIARY RESULT 260 3.2 ESTIMATES OF THE
HAUSDORFF MEASURE AND OF HAUSDORFF DIMENSION 262 3.3 THE GENERALIZED
BENDIXSON CRITERION 266 3.4 ON THE FINITENESS OF THE NUMBER OF PERIODIC
SOLUTIONS . . 267 3.5 CONVERGENCE THEOREMS 268 4 CONVERGENCE IN
THIRD-ORDER NONLINEAR SYSTEMS 269 4.1 THE GENERALIZED LORENZ SYSTEM 269
4.2 EULER S EQUATIONS DESCRIBING THE ROTATION OF A RIGID BODY IN A
RESISTING MEDIUM 275 4.3 A NONLINEAR SYSTEM ARISING FROM FLUID
CONVECTION IN A ROTATING ELLIPSOID 276 4.4 A SYSTEM DESCRIBING THE
INTERACTION OF THREE WAVES IN PLASMA 277 5 ESTIMATES OF FRACTAL
DIMENSION 280 5.1 MAPS WITH A CONSTANT JACOBIAN 280 12 CONTENTS 5.2
AUTONOMOUS DIFFERENTIAL EQUATIONS WHICH ARE CONSERVA- TIVE ON THE
INVARIANT SET 283 6 ESTIMATES OF THE TOPOLOGICAL ENTROPY 285 6.1 ITO S
GENERALIZED ENTROPY ESTIMATE FOR MAPS 285 6.2 APPLICATION TO
DIFFERENTIAL EQUATIONS 290 7 FRACTAL DIMENSION ESTIMATES 292 7.1 THE
ROSSLER SYSTEM 293 7.2 LORENZ EQUATION 295 7.3 EQUATIONS OF THE THIRD
ORDER 302 7.4 EQUATIONS DESCRIBING THE INTERACTION BETWEEN WAVES IN
PLASMA 307 8 UPPER LYAPUNOV DIMENSION 310 8.1 DEFINITION OF LOCAL
LYAPUNOV EXPONENTS 310 8.2 AN UPPER ESTIMATE FOR THE UPPER LYAPUNOV
DIMENSION OF THE ATTRACTORS OF THE LORENZ SYSTEM 313 9 FORMULAS FOR THE
LYAPUNOV DIMENSION 318 9.1 GENERAL RESULTS 318 9.2 THE HENON MAP 325 9.3
LORENZ SYSTEM 328 10 INVARIANT SETS OF VECTOR FIELDS 331 10.1
INTRODUCTION 331 10.2 HAUSDORFF DIMENSION BOUNDS FOR INVARIANT SETS OF
MAPS ON MANIFOLDS 332 10.3 TIME-DEPENDENT VECTOR FIELDS ON MANIFOLDS 337
10.4 CONVERGENCE FOR AUTONOMOUS VECTOR FIELDS 343 11 USE OF A TUBULAR
CARATHEODORY STRUCTURE 346 11.1 THE SYSTEM IN NORMAL VARIATION 346 11.2
TUBULAR CARATHEODORY STRUCTURE 350 11.3 DIMENSION ESTIMATES FOR SETS
WHICH ARE NEGATIVELY INVARIANT FOR A FLOW 352 11.4 FLOW INVARIANT SETS
WITH AN EQUIVARIANT TANGENT BUNDLE SPLITTING 359 11.5 GENERALIZATIONS OF
THE THEOREMS OF HARTMAN-OLECH AND BORG 362 12 THE LYAPUNOV DIMENSION AS
UPPER BOUND 364 12.1 STATEMENT OF THE RESULTS 364 12.2 PROOF OF THEOREM
12.1.1 365 12.3 GLOBAL LYAPUNOV EXPONENTS AND UPPER LYAPUNOV DIMENSION
372 12.4 APPLICATION TO THE LORENZ SYSTEM 373 CONTENTS 13 13 LOWER
ESTIMATES OF THE DIMENSION OF B- ATTRACTORS 376 13.1 INTRODUCTION 376
13.2 FREQUENCY-DOMAIN CONDITIONS FOR LOWER TOPOLOGICAL DIMENSION BOUNDS
OF GLOBAL #-ATTRACTORS 376 13.3 LOWER ESTIMATES OF THE HAUSDORFF
DIMENSION OF GLOBAL B-ATTRACTORS 381 13.4 LOWER DIMENSION ESTIMATES FOR
GLOBAL ATTRACTORS BASED ON THE EVOLUTION OF CURRENTS 382 A SOME TOOLS
385 A.I DEFINITION OF A DIFFERENTIABLE MANIFOLD 385 A.2 TANGENT SPACE,
TANGENT BUNDLE AND DIFFERENTIAL 387 A.3 TENSOR PRODUCTS, EXTERIOR
PRODUCTS AND TENSOR FIELDS . . . 389 A.4 RIEMANNIAN MANIFOLDS 390 A.5
COVARIANT DERIVATIVE 392 A.6 VECTOR FIELDS 392 A.7 SPACES OF VECTOR
FIELDS AND MAPS 394 A.8 PARALLEL TRANSPORT, GEODESIES AND EXPONENTIAL
MAP . . . . 397 A.9 CURVATURE AND TORSION 398 A.10 FIBER BUNDLES AND
DISTRIBUTIONS 400 A.LL RECURRENCE AND HYPERBOLICITY IN DYNAMICAL SYSTEMS
. . . 402 A.12 HOMOLOGY THEORY 403 A.13 DEGREE THEORY 405 A.14 SIMPLE
5-LINKED PARAMETERIZED M-BOUNDARIES 407 A.15 GEOMETRIC MEASURE THEORY
409 A.16 TOTALLY ORDERED SETS 411 A.17 ALMOST PERIODIC FUNCTIONS 412
BIBLIOGRAPHY 415 INDEX 435
|
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| author_facet | Bojčenko, Vladimir A. 1952- Leonov, Gennadij A. 1947- Reitmann, Volker 1948- |
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| bvnumber | BV017051993 |
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| discipline | Mathematik |
| edition | 1. ed. |
| format | Book |
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| id | DE-604.BV017051993 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:13:16Z |
| institution | BVB |
| isbn | 3519004372 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010292212 |
| oclc_num | 181434280 |
| open_access_boolean | |
| owner | DE-384 DE-898 DE-BY-UBR DE-824 DE-19 DE-BY-UBM DE-634 DE-11 DE-188 DE-739 |
| owner_facet | DE-384 DE-898 DE-BY-UBR DE-824 DE-19 DE-BY-UBM DE-634 DE-11 DE-188 DE-739 |
| physical | 441 S. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Teubner |
| record_format | marc |
| series | Teubner-Texte zur Mathematik |
| series2 | Teubner-Texte zur Mathematik |
| spelling | Bojčenko, Vladimir A. 1952- Verfasser (DE-588)115047832 aut Dimension theory for ordinary differential equations Vladimir A. Boichenko ; Gennadij A. Leonov ; Volker Reitmann 1. ed. Wiesbaden Teubner 2005 441 S. txt rdacontent n rdamedia nc rdacarrier Teubner-Texte zur Mathematik 141 Ljapunov-Funktion (DE-588)4274502-0 gnd rswk-swf Dimensionstheorie (DE-588)4149935-9 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Dimensionstheorie (DE-588)4149935-9 s Ljapunov-Funktion (DE-588)4274502-0 s DE-604 Leonov, Gennadij A. 1947- Verfasser (DE-588)111346487 aut Reitmann, Volker 1948- Verfasser (DE-588)111346495 aut Teubner-Texte zur Mathematik 141 (DE-604)BV000012607 141 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010292212&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bojčenko, Vladimir A. 1952- Leonov, Gennadij A. 1947- Reitmann, Volker 1948- Dimension theory for ordinary differential equations Teubner-Texte zur Mathematik Ljapunov-Funktion (DE-588)4274502-0 gnd Dimensionstheorie (DE-588)4149935-9 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4274502-0 (DE-588)4149935-9 (DE-588)4020929-5 |
| title | Dimension theory for ordinary differential equations |
| title_auth | Dimension theory for ordinary differential equations |
| title_exact_search | Dimension theory for ordinary differential equations |
| title_full | Dimension theory for ordinary differential equations Vladimir A. Boichenko ; Gennadij A. Leonov ; Volker Reitmann |
| title_fullStr | Dimension theory for ordinary differential equations Vladimir A. Boichenko ; Gennadij A. Leonov ; Volker Reitmann |
| title_full_unstemmed | Dimension theory for ordinary differential equations Vladimir A. Boichenko ; Gennadij A. Leonov ; Volker Reitmann |
| title_short | Dimension theory for ordinary differential equations |
| title_sort | dimension theory for ordinary differential equations |
| topic | Ljapunov-Funktion (DE-588)4274502-0 gnd Dimensionstheorie (DE-588)4149935-9 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Ljapunov-Funktion Dimensionstheorie Gewöhnliche Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010292212&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000012607 |
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