Ordinary differential equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Berlin [u.a.]
Springer
1992
|
| Edition: | 3. ed. |
| Series: | Springer-Textbook
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | EST: Obyknovennye differencial'nye uravnenija <engl.> |
| Physical Description: | 334 S. graph. Darst. |
| ISBN: | 3540548130 0387548130 |
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Record in the Search Index
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| adam_text | VLADIMIR I. ARNOL D ORDINARY DIFFERENTIAL EQUATIONS TRANSLATED FROM THE
RUSSIAN BY ROGER COOKE WITH 272 FIGURES SPRINGER-VERLAG BERLIN
HEIDELBERG NEW YORK LONDON PARIS TOKYO HONG KONG BARCELONA BUDAPEST
CONTENTS CHAPTER 1. BASIC CONCEPTS 13 § 1. PHASE SPACES 13 1. EXAMPLES
OF EVOLUTIONARY PROCESSES 13 2. PHASE SPACES 14 3. THE INTEGRAL CURVES
OF A DIRECTION FIELD 16 4. A DIFFERENTIAL EQUATION AND ITS SOLUTIONS 17
5. THE EVOLUTIONARY EQUATION WITH A ONE-DIMENSIONAL PHASE SPACE 19 6.
EXAMPLE: THE EQUATION OF NORMAL REPRODUCTION 21 7. EXAMPLE: THE
EXPLOSION EQUATION 23 8. EXAMPLE: THE LOGISTIC CURVE 24 9. EXAMPLE:
HARVEST QUOTAS 25 10. EXAMPLE: HARVESTING WITH A RELATIVE QUOTA 26 11.
EQUATIONS WITH A MULTIDIMENSIONAL PHASE SPACE 27 12. EXAMPLE: THE
DIFFERENTIAL EQUATION OF A PREDATOR-PREY SYSTEM 28 13. EXAMPLE: A FREE
PARTICLE ON A LINE 31 14. EXAMPLE: FREE FALL 32 15. EXAMPLE: SMALL
OSCILLATIONS 32 16. EXAMPLE: THE MATHEMATICAL PENDULUM 33 17. EXAMPLE:
THE INVERTED PENDULUM 34 18. EXAMPLE: SMALL OSCILLATIONS OF A SPHERICAL
PENDULUM 34 § 2. VECTOR FIELDS ON THE LINE 36 1. EXISTENCE AND
UNIQUENESS OF SOLUTIONS 36 2. A COUNTEREXAMPLE 36 3. PROOF OF UNIQUENESS
37 4. DIRECT PRODUCTS 39 5. EXAMPLES OF DIRECT PRODUCTS 39 6. EQUATIONS
WITH SEPARABLE VARIABLES 41 7. AN EXAMPLE: THE LOTKA-VOLTERRA MODEL 43 §
3. LINEAR EQUATIONS 48 1. HOMOGENEOUS LINEAR EQUATIONS 48 CONTENTS 7 2.
FIRST-ORDER HOMOGENEOUS LINEAR EQUATIONS WITH PERIODIC COEFFICIENTS 49
3. INHOMOGENEOUS LINEAR EQUATIONS 51 4. THE INFLUENCE FUNCTION AND
5-SHAPED INHOMOGENEITIES 53 5. INHOMOGENEOUS LINEAR EQUATIONS WITH
PERIODIC COEFFICIENTS .... 56 § 4. PHASE FLOWS 57 1. THE ACTION OF A
GROUP ON A SET 57 2. ONE-PARAMETER TRANSFORMATION GROUPS 59 3.
ONE-PARAMETER DIFFEOMORPHISM GROUPS 61 4. THE PHASE VELOCITY VECTOR
FIELD 63 § 5. THE ACTION OF DIFFEOMORPHISMS ON VECTOR FIELDS AND
DIRECTION FIELDS 66 1. THE ACTION OF SMOOTH MAPPINGS ON VECTORS 66 2.
THE ACTION OF DIFFEOMORPHISMS ON VECTOR FIELDS 70 3. CHANGE OF VARIABLES
IN AN EQUATION 72 4. THE ACTION OF A DIFFEOMORPHISM ON A DIRECTION FIELD
73 5. THE ACTION OF A DIFFEOMORPHISM ON A PHASE FLOW 75 § 6. SYMMETRIES
76 1. SYMMETRY GROUPS 76 2. APPLICATION OF A ONE-PARAMETER SYMMETRY
GROUP TO INTEGRATE AN EQUATION 77 3. HOMOGENEOUS EQUATIONS 79 4.
QUASI-HOMOGENEOUS EQUATIONS 82 5. SIMILARITY AND DIMENSIONAL
CONSIDERATIONS 84 6. METHODS OF INTEGRATING DIFFERENTIAL EQUATIONS 86
CHAPTER 2. BASIC THEOREMS 89 § 7. RECTIFICATION THEOREMS 89 1.
RECTIFICATION OF A DIRECTION FIELD 89 2. EXISTENCE AND UNIQUENESS
THEOREMS 92 3. THEOREMS ON CONTINUOUS AND DIFFERENTIABLE DEPENDENCE OF
THE SOLUTIONS ON THE INITIAL CONDITION 93 4. TRANSFORMATION OVER THE
TIME INTERVAL FROM T 0 TO T 96 5. THEOREMS ON CONTINUOUS AND
DIFFERENTIABLE DEPENDENCE ON A PARAMETER 97 6. EXTENSION THEOREMS 100 7.
RECTIFICATION OF A VECTOR FIELD 103 § 8. APPLICATIONS TO EQUATIONS OF
HIGHER ORDER THAN FIRST .... 104 1. THE EQUIVALENCE OF AN EQUATION OF
ORDER N AND A SYSTEM OF N FIRST-ORDER EQUATIONS 104 2. EXISTENCE AND
UNIQUENESS THEOREMS 107 3. DIFFERENTIABILITY AND EXTENSION THEOREMS 108
8 CONTENTS 4. SYSTEMS OF EQUATIONS 109 5. REMARKS ON TERMINOLOGY 112 §
9. THE PHASE CURVES OF AN AUTONOMOUS SYSTEM 116 1. AUTONOMOUS SYSTEMS
117 2. TRANSLATION OVER TIME 117 3. CLOSED PHASE CURVES 119 § 10 . THE
DERIVATIVE IN THE DIRECTION OF A VECTOR FIELD AND FIRST INTEGRALS 121 1.
THE DERIVATIVE IN THE DIRECTION OF A VECTOR 121 2. THE DERIVATIVE IN THE
DIRECTION OF A VECTOR FIELD 122 3. PROPERTIES OF THE DIRECTIONAL
DERIVATIVE 123 4. THE LIE ALGEBRA OF VECTOR FIELDS 124 5. FIRST
INTEGRALS .. .,-.* 125 6. LOCAL FIRST INTEGRALS 126 7. TIME-DEPENDENT
FIRST INTEGRALS 127 § 11. FIRST-ORDER LINEAR AND QUASI-LINEAR PARTIAL
DIFFERENTIAL EQUATIONS 129 1. THE HOMOGENEOUS LINEAR EQUATION 129 2. THE
CAUCHY PROBLEM 130 3. THE INHOMOGENEOUS LINEAR EQUATION 131 4. THE
QUASI-LINEAR EQUATION 132 5. THE CHARACTERISTICS OF A QUASI-LINEAR
EQUATION 133 6. INTEGRATION OF A QUASI-LINEAR EQUATION 135 7. THE
FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATION 136 § 12. THE
CONSERVATIVE SYSTEM WITH ONE DEGREE OF FREEDOM.. 138 1. DEFINITIONS 138
2. THE LAW OF CONSERVATION OF ENERGY 139 3. THE LEVEL LINES OF THE
ENERGY 140 4. THE LEVEL LINES OF THE ENERGY NEAR A SINGULAR POINT 142 5.
EXTENSION OF THE SOLUTIONS OF NEWTON S EQUATION 144 6. NONCRITICAL LEVEL
LINES OF THE ENERGY 145 7. PROOF OF THE THEOREM OF SECT. 6 146 8.
CRITICAL LEVEL LINES 147 9. AN EXAMPLE 148 10. SMALL PERTURBATIONS OF A
CONSERVATIVE SYSTEM 149 CHAPTER 3. LINEAR SYSTEMS 152 § 13. LINEAR
PROBLEMS 152 1. EXAMPLE: LINEARIZATION 152 2. EXAMPLE: ONE-PARAMETER
GROUPS OF LINEAR TRANSFORMATIONS OF R N 153 3. THE LINEAR EQUATION 154 §
14. THE EXPONENTIAL FUNCTION 155 CONTENTS 9 1. THE NORM OF AN OPERATOR
155 2. THE METRIC SPACE OF OPERATORS 156 3. PROOF OF COMPLETENESS 156 4.
SERIES 157 5. DEFINITION OF THE EXPONENTIAL E A 158 6. AN EXAMPLE 159 7.
THE EXPONENTIAL OF A DIAGONAL OPERATOR 160 8. THE EXPONENTIAL OF A
NILPOTENT OPERATOR 160 9. QUASI-POLYNOMIALS 161 § 15. PROPERTIES OF THE
EXPONENTIAL 162 1. THE GROUP PROPERTY 163 2. THE FUNDAMENTAL THEOREM OF
THE THEORY OF LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 164 3. THE
GENERAL FORM OF ONE-PARAMETER GROUPS OF LINEAR TRANSFORMATIONS OF THE
SPACE R N 165 4. A SECOND DEFINITION OF THE EXPONENTIAL 165 5. AN
EXAMPLE: EULER S FORMULA, FOR E Z 166 6. EULER S BROKEN LINES 167 § 16.
THE DETERMINANT OF AN EXPONENTIAL 169 1. THE DETERMINANT OF AN OPERATOR
169 2. THE TRACE OF AN OPERATOR 170 3. THE CONNECTION BETWEEN THE
DETERMINANT AND THE TRACE 171 4. THE DETERMINANT OF THE OPERATOR E A 171
§ 17. PRACTICAL COMPUTATION OF THE MATRIX OF AN EXPONENTIAL * THE CASE
WHEN THE EIGENVALUES ARE REAL AND DISTINCT .. 173 1. THE DIAGONALIZABLE
OPERATOR 173 2. AN EXAMPLE 174 3. THE DISCRETE CASE 175 § 18.
COMPLEXIFICATION AND REALIFICATION 177 1. REALIFICATION 177 2.
COMPLEXIFICATION 177 3. THE COMPLEX CONJUGATE 178 4. THE EXPONENTIAL,
DETERMINANT, AND TRACE OF A COMPLEX OPERATOR 179 5. THE DERIVATIVE OF A
CURVE WITH COMPLEX VALUES 180 § 19. THE LINEAR EQUATION WITH A COMPLEX
PHASE SPACE 181 1. DEFINITIONS 181 2. THE FUNDAMENTAL THEOREM 181 3. THE
DIAGONALIZABLE CASE 182 4. EXAMPLE: A LINEAR EQUATION WHOSE PHASE SPACE
IS A COMPLEX LINE 182 5. COROLLARY 185 § 20. THE COMPLEXIFICATION OF A
REAL LINEAR EQUATION 185 1. THE COMPLEXIFIED EQUATION 185 10 CONTENTS 2.
THE INVARIANT SUBSPACES OF A REAL OPERATOR 187 3. THE LINEAR EQUATION ON
THE PLANE 189 4. THE CLASSIFICATION OF SINGULAR POINTS IN THE PLANE 190
5. EXAMPLE: THE PENDULUM WITH FRICTION 191 6. THE GENERAL SOLUTION OF A
LINEAR EQUATION IN THE CASE WHEN THE CHARACTERISTIC EQUATION HAS ONLY
SIMPLE ROOTS 193 § 21. THE CLASSIFICATION OF SINGULAR POINTS OF LINEAR
SYSTEMS. 195 1. EXAMPLE: SINGULAR POINTS IN THREE-DIMENSIONAL SPACE 195
2. LINEAR, DIFFERENTIABLE, AND TOPOLOGICAL EQUIVALENCE 197 3. THE LINEAR
CLASSIFICATION 198 4. THE DIFFERENTIABLE CLASSIFICATION 199 § 22. THE
TOPOLOGICAL CLASSIFICATION OF SINGULAR POINTS 199 1. THEOREM 199 2.
REDUCTION TO THE CASE M_ = 0 200 3. THE LYAPUNOV FUNCTION 201 4.
CONSTRUCTION OF THE LYAPUNOV FUNCTION 202 5. AN ESTIMATE OF THE
DERIVATIVE 204 6. CONSTRUCTION OF THE HOMEOMORPHISM H 206 7. PROOF OF
LEMMA 3 207 8. PROOF OF THE TOPOLOGICAL CLASSIFICATION THEOREM 208 § 23.
STABILITY OF EQUILIBRIUM POSITIONS 210 1. LYAPUNOV STABILITY 210 2.
ASYMPTOTIC STABILITY 211 3. A THEOREM ON STABILITY IN FIRST
APPROXIMATION 211 4. PROOF OF THE THEOREM 212 § 24. THE CASE OF PURELY
IMAGINARY EIGENVALUES 215 1. THE TOPOLOGICAL CLASSIFICATION 215 2. AN
EXAMPLE 215 3. THE PHASE CURVES OF EQ. (4) ON THE TORUS 217 4.
COROLLARIES 219 5. THE MULTIDIMENSIONAL CASE 219 6. THE UNIFORM
DISTRIBUTION 220 § 25. THE CASE OF MULTIPLE EIGENVALUES 221 1. THE
COMPUTATION OF E A T, WHERE A IS A JORDAN BLOCK 221 2. APPLICATIONS 223
3. APPLICATIONS TO SYSTEMS OF EQUATIONS OF ORDER HIGHER THAN THE FIRST
224 4. THE CASE OF A SINGLE ??TH-ORDER EQUATION 225 5. ON RECURSIVE
SEQUENCES 226 6. SMALL OSCILLATIONS 227 § 26. QUASI-POLYNOMIALS 229 1. A
LINEAR FUNCTION SPACE 229 2. THE VECTOR SPACE OF SOLUTIONS OF A LINEAR
EQUATION 230 CONTENTS 11 3. TRANSLATION-INVARIANCE 231 4. HISTORICAL
REMARK 232 5. INHOMOGENEOUS EQUATIONS 233 6. THE METHOD OF COMPLEX
AMPLITUDES 235 7. APPLICATION TO THE CALCULATION OF WEAKLY NONLINEAR
OSCILLATIONS 240 § 27. NONAUTONOMOUS LINEAR EQUATIONS 241 1. DEFINITION
241 2. THE EXISTENCE OF SOLUTIONS 242 3. THE VECTOR SPACE OF SOLUTIONS
244 4. THE WRONSKIAN DETERMINANT 245 5. THE CASE OF A SINGLE EQUATION
246 6. LIOUVILLE S THEOREM 248 7. STURM S THEOREMS ON THE ZEROS OF
SOLUTIONS OF SECOND-ORDER EQUATIONS 251 § 28. LINEAR EQUATIONS WITH
PERIODIC COEFFICIENTS 256 1. THE MAPPING OVER A PERIOD 256 2. STABILITY
CONDITIONS 258 3. STRONGLY STABLE SYSTEMS 259 4. COMPUTATIONS 262 § 29.
VARIATION OF CONSTANTS 26 4 1. THE SIMPLEST CASE 264 2. THE GENERAL CASE
264 3. COMPUTATIONS 265 CHAPTER 4. PROOFS OF THE MAIN THEOREMS 267 § 30.
CONTRACTION MAPPINGS 267 1. DEFINITION 267 2. THE CONTRACTION MAPPING
THEOREM 268 3. REMARK 269 § 31. PROOF OF THE THEOREMS ON EXISTENCE AND
CONTINUOUS DEPENDENCE ON THE INITIAL CONDITIONS 269 1. THE SUCCESSIVE
APPROXIMATIONS OF PICARD 269 2. PRELIMINARY ESTIMATES 271 3. THE
LIPSCHITZ CONDITION 272 4. DIFFERENTIABILITY AND THE LIPSCHITZ CONDITION
272 5. THE QUANTITIES C, L,A ,B 273 6. THE METRIC SPACE M 274 7. THE
CONTRACTION MAPPING A : M -+ M 275 8. THE EXISTENCE AND UNIQUENESS
THEOREM 276 9. OTHER APPLICATIONS OF CONTRACTION MAPPINGS 277 § 32. THE
THEOREM ON DIFFERENTIABILITY 279 1. THE EQUATION OF VARIATIONS 279 2.
THE DIFFERENTIABILITY THEOREM 280 12 CONTENTS 3. HIGHER DERIVATIVES WITH
RESPECT TO X 281 4. DERIVATIVES IN X AND T 28 1 5. THE RECTIFICATION
THEOREM 282 6. THE LAST DERIVATIVE 285 CHAPTER 5. DIFFERENTIAL EQUATIONS
ON MANIFOLDS 288 § 33. DIFFERENTIABLE MANIFOLDS 288 1. EXAMPLES OF
MANIFOLDS 288 2. DEFINITIONS 288 3. EXAMPLES OF ATLASES 291 4.
COMPACTNESS 293 5. CONNECTEDNESS AND DIMENSION 293 6. DIFFERENTIABLE
MAPPINGS 294 7. REMARK 296 8. SUBMANIFOLDS 296 9. AN EXAMPLE 297 § 34.
THE TANGENT BUNDLE. VECTOR FIELDS ON A MANIFOLD 298 1. THE TANGENT SPACE
298 2. THE TANGENT BUNDLE 299 3. A REMARK ON PARALLELIZABILITY 301 4.
THE TANGENT MAPPING 302 5. VECTOR FIELDS 303 § 35. THE PHASE FLOW
DEFINED BY A VECTOR FIELD 304 1. THEOREM 304 2. CONSTRUCTION OF THE
DIFFEOMORPHISMS G L FOR SMALL T 305 3. THE CONSTRUCTION OF / FOR ANY T
306 4. A REMARK 307 § 36. THE INDICES OF THE SINGULAR POINTS OF A VECTOR
FIELD 309 1. THE INDEX OF A. CURVE 309 2. PROPERTIES OF THE INDEX 310 3.
EXAMPLES 310 4. THE INDEX OF A SINGULAR POINT OF A VECTOR FIELD 312 5.
THE THEOREM ON THE SUM OF THE INDICES 313 6. THE SUM OF THE INDICES OF
THE SINGULAR POINTS ON A SPHERE 315 7. JUSTIFICATION 317 8. THE
MULTIDIMENSIONAL CASE 318 EXAMINATION TOPICS 323 SAMPLE EXAMINATION
PROBLEMS 324 SUPPLEMENTARY PROBLEMS 326 SUBJECT INDEX 331
|
| any_adam_object | 1 |
| author | Arnolʹd, V. I. 1937-2010 |
| author_GND | (DE-588)119540878 |
| author_facet | Arnolʹd, V. I. 1937-2010 |
| author_role | aut |
| author_sort | Arnolʹd, V. I. 1937-2010 |
| author_variant | v i a vi via |
| building | Verbundindex |
| bvnumber | BV008265046 |
| callnumber-first | Q - Science |
| callnumber-label | QA372 |
| callnumber-raw | QA372 |
| callnumber-search | QA372 |
| callnumber-sort | QA 3372 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 520 |
| ctrlnum | (OCoLC)25026371 (DE-599)BVBBV008265046 |
| dewey-full | 515/.352 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.352 |
| dewey-search | 515/.352 |
| dewey-sort | 3515 3352 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| genre | 1\p (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV008265046 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:17:24Z |
| institution | BVB |
| isbn | 3540548130 0387548130 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005459270 |
| oclc_num | 25026371 |
| open_access_boolean | |
| owner | DE-703 DE-824 |
| owner_facet | DE-703 DE-824 |
| physical | 334 S. graph. Darst. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer-Textbook |
| spelling | Arnolʹd, V. I. 1937-2010 Verfasser (DE-588)119540878 aut Ordinary differential equations Vladimir I. Arnol'd 3. ed. Berlin [u.a.] Springer 1992 334 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer-Textbook EST: Obyknovennye differencial'nye uravnenija <engl.> Equations différentielles ram Gewone differentiaalvergelijkingen gtt Équations différentielles Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Lineares System (DE-588)4125617-7 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 Differentialgleichung (DE-588)4012249-9 s 2\p DE-604 Lineares System (DE-588)4125617-7 s 3\p DE-604 Mannigfaltigkeit (DE-588)4037379-4 s 4\p DE-604 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005459270&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Arnolʹd, V. I. 1937-2010 Ordinary differential equations Equations différentielles ram Gewone differentiaalvergelijkingen gtt Équations différentielles Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Differentialgleichung (DE-588)4012249-9 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Lineares System (DE-588)4125617-7 gnd |
| subject_GND | (DE-588)4020929-5 (DE-588)4012249-9 (DE-588)4037379-4 (DE-588)4125617-7 (DE-588)4123623-3 |
| title | Ordinary differential equations |
| title_auth | Ordinary differential equations |
| title_exact_search | Ordinary differential equations |
| title_full | Ordinary differential equations Vladimir I. Arnol'd |
| title_fullStr | Ordinary differential equations Vladimir I. Arnol'd |
| title_full_unstemmed | Ordinary differential equations Vladimir I. Arnol'd |
| title_short | Ordinary differential equations |
| title_sort | ordinary differential equations |
| topic | Equations différentielles ram Gewone differentiaalvergelijkingen gtt Équations différentielles Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Differentialgleichung (DE-588)4012249-9 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Lineares System (DE-588)4125617-7 gnd |
| topic_facet | Equations différentielles Gewone differentiaalvergelijkingen Équations différentielles Differential equations Gewöhnliche Differentialgleichung Differentialgleichung Mannigfaltigkeit Lineares System Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005459270&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT arnolʹdvi ordinarydifferentialequations |