Approaches to the qualitative theory of ordinary differential equations: dynamical systems and nonlinear oscillations
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2007
|
| Series: | Peking University series in mathematics
3 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz.: S. 377 - 383 |
| Physical Description: | IX, 383 S. graph. Darst. |
| ISBN: | 981270468X 9789812704689 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface v
Chapter 1. Cauchy Problem 1
1.1 Fundamental Theorems 1
1.2 Method of Euler Polygons 15
1.3 Local Behavior of Integral Curves 20
1.4 Peano Phenomenon 24
1.5 Convergence Theorem on Difference Methods 33
Chapter 2. Global Behavior of Solution 47
2.1 Global Existence of Solution 47
2.2 Predictability of Solution 58
2.3 Liapunov Stability 61
2.4 Liapunov Unstability 71
Chapter 3. Autonomous Systems 73
3.1 Phase Portrait 73
3.2 Orbital Box 76
3.3 Types of Orbits 77
3.4 Singular Points 79
3.5 General Property of Singular Points 86
3.6 Closed Orbit 87
3.7 Invariant Torus 91
3.8 Limit-Point Set 95
3.9 Poincaré-Bendixson Theorem 97
viii Contents
Chapter 4. Non-Autonomous Systems 101
4.1 General Systems 101
4.2 Conservative Systems 104
4.3 Dissipative Systems 106
4.4 Planar Periodic Systems 115
4.5 Invariant Continuum 119
Chapter 5. Dynamical Systems 123
5.1 The Originality 123
5.2 Recurrence 128
5.3 Quasi-Minimal Set 132
5.4 Minimal Set 134
5.5 Almost Periodic Motion 144
Chapter 6. Fixed-Point Theorems 155
6.1 Poincaré Index 155
6.2 Vector Fields on Closed Surfaces 165
6.3 Spatial Vector Fields 172
6.4 Fixed-Point Theorems of Brouwer Type 176
Chapter 7. Bend-Twist Theorem 181
7.1 Generalized Poincaré-Birkhoff Twist Theorem 181
7.2 Analytic Bend-Twist Theorem 184
7.3 Analytic Poincaré-Birkhoff Twist Theorem 189
7.4 Application of the Bend-Twist Theorem 191
Chapter 8. Chaotic Motions 199
8.1 Definition of Chaotic Motion 199
8.2 Chaotic Quasi-Minimal Set 201
8.3 Sufficient Conditions for Chaotic Sets 202
8.4 Chaotic Closed Surfaces 205
8.5 Applications 209
Chapter 9. Perturbation Method 217
9.1 Nonlinear Differential Equation of Second Order 217
9.2 Method of Averaging 225
9.3 High Frequency Forced Oscillations 230
Contents ¡x
Chapter 10. Duffing Equations of Second Order 241
10.1 Periodic Oscillations 241
10.2 Time-Map 250
10.3 Duffing Equation of Super-Linear Type 261
10.4 Duffing Equation of Sub-Linear Type 275
10.5 Duffing Equation of Semi-Linear Type 288
Chapter 11. Some Special Problems 313
11.1 Reeb s Problem 313
11.2 Birkhoff s Conjecture 319
11.3 Morse s Conjecture 326
11.4 Kolmogorov s Problem 331
11.5 Brillouin Focusing System 345
11.6 A Retarded Equation 355
11.7 Periodic Lotka-Volterra System 365
Bibliography 377
|
| adam_txt |
Contents
Preface v
Chapter 1. Cauchy Problem 1
1.1 Fundamental Theorems 1
1.2 Method of Euler Polygons 15
1.3 Local Behavior of Integral Curves 20
1.4 Peano Phenomenon 24
1.5 Convergence Theorem on Difference Methods 33
Chapter 2. Global Behavior of Solution 47
2.1 Global Existence of Solution 47
2.2 Predictability of Solution 58
2.3 Liapunov Stability 61
2.4 Liapunov Unstability 71
Chapter 3. Autonomous Systems 73
3.1 Phase Portrait 73
3.2 Orbital Box 76
3.3 Types of Orbits 77
3.4 Singular Points 79
3.5 General Property of Singular Points 86
3.6 Closed Orbit 87
3.7 Invariant Torus 91
3.8 Limit-Point Set 95
3.9 Poincaré-Bendixson Theorem 97
viii Contents
Chapter 4. Non-Autonomous Systems 101
4.1 General Systems 101
4.2 Conservative Systems 104
4.3 Dissipative Systems 106
4.4 Planar Periodic Systems 115
4.5 Invariant Continuum 119
Chapter 5. Dynamical Systems 123
5.1 The Originality 123
5.2 Recurrence 128
5.3 Quasi-Minimal Set 132
5.4 Minimal Set 134
5.5 Almost Periodic Motion 144
Chapter 6. Fixed-Point Theorems 155
6.1 Poincaré Index 155
6.2 Vector Fields on Closed Surfaces 165
6.3 Spatial Vector Fields 172
6.4 Fixed-Point Theorems of Brouwer Type 176
Chapter 7. Bend-Twist Theorem 181
7.1 Generalized Poincaré-Birkhoff Twist Theorem 181
7.2 Analytic Bend-Twist Theorem 184
7.3 Analytic Poincaré-Birkhoff Twist Theorem 189
7.4 Application of the Bend-Twist Theorem 191
Chapter 8. Chaotic Motions 199
8.1 Definition of Chaotic Motion 199
8.2 Chaotic Quasi-Minimal Set 201
8.3 Sufficient Conditions for Chaotic Sets 202
8.4 Chaotic Closed Surfaces 205
8.5 Applications 209
Chapter 9. Perturbation Method 217
9.1 Nonlinear Differential Equation of Second Order 217
9.2 Method of Averaging 225
9.3 High Frequency Forced Oscillations 230
Contents ¡x
Chapter 10. Duffing Equations of Second Order 241
10.1 Periodic Oscillations 241
10.2 Time-Map 250
10.3 Duffing Equation of Super-Linear Type 261
10.4 Duffing Equation of Sub-Linear Type 275
10.5 Duffing Equation of Semi-Linear Type 288
Chapter 11. Some Special Problems 313
11.1 Reeb's Problem 313
11.2 Birkhoff's Conjecture 319
11.3 Morse's Conjecture 326
11.4 Kolmogorov's Problem 331
11.5 Brillouin Focusing System 345
11.6 A Retarded Equation 355
11.7 Periodic Lotka-Volterra System 365
Bibliography 377 |
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| index_date | 2024-07-02T20:04:55Z |
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| isbn | 981270468X 9789812704689 |
| language | English |
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| spelling | Ding, Tongren Verfasser aut Approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations Ding Tongren New Jersey [u.a.] World Scientific 2007 IX, 383 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Peking University series in mathematics 3 Literaturverz.: S. 377 - 383 Equações diferenciais ordinárias (análise;textos avançados) larpcal Sistemas dinâmicos larpcal Differentiable dynamical systems Textbooks Differential equations, Nonlinear Textbooks Nonlinear oscillations Textbooks Anharmonischer Oszillator (DE-588)4421420-0 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s Dynamisches System (DE-588)4013396-5 s Anharmonischer Oszillator (DE-588)4421420-0 s DE-604 Peking University series in mathematics 3 (DE-604)BV021821556 3 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016378003&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ding, Tongren Approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations Peking University series in mathematics Equações diferenciais ordinárias (análise;textos avançados) larpcal Sistemas dinâmicos larpcal Differentiable dynamical systems Textbooks Differential equations, Nonlinear Textbooks Nonlinear oscillations Textbooks Anharmonischer Oszillator (DE-588)4421420-0 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4421420-0 (DE-588)4020929-5 (DE-588)4013396-5 |
| title | Approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations |
| title_auth | Approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations |
| title_exact_search | Approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations |
| title_exact_search_txtP | Approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations |
| title_full | Approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations Ding Tongren |
| title_fullStr | Approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations Ding Tongren |
| title_full_unstemmed | Approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations Ding Tongren |
| title_short | Approaches to the qualitative theory of ordinary differential equations |
| title_sort | approaches to the qualitative theory of ordinary differential equations dynamical systems and nonlinear oscillations |
| title_sub | dynamical systems and nonlinear oscillations |
| topic | Equações diferenciais ordinárias (análise;textos avançados) larpcal Sistemas dinâmicos larpcal Differentiable dynamical systems Textbooks Differential equations, Nonlinear Textbooks Nonlinear oscillations Textbooks Anharmonischer Oszillator (DE-588)4421420-0 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Equações diferenciais ordinárias (análise;textos avançados) Sistemas dinâmicos Differentiable dynamical systems Textbooks Differential equations, Nonlinear Textbooks Nonlinear oscillations Textbooks Anharmonischer Oszillator Gewöhnliche Differentialgleichung Dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016378003&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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| work_keys_str_mv | AT dingtongren approachestothequalitativetheoryofordinarydifferentialequationsdynamicalsystemsandnonlinearoscillations |