Nonlinear ordinary differential equations:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Clarendon Press
1987
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| Edition: | 2. ed. |
| Series: | Oxford applied mathematics and computing science series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 381 S. graph. Darst. |
| ISBN: | 019859657X 0198596561 |
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MARC
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| 245 | 1 | 0 | |a Nonlinear ordinary differential equations |c D. W. Jordan and P. Smith |
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| 650 | 4 | |a Équations différentielles non linéaires | |
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Record in the Search Index
| _version_ | 1804116744225161216 |
|---|---|
| adam_text | Titel: Nonlinear ordinary differential equations
Autor: Jordan, Dominic W
Jahr: 1987
Contents 1. SECOND-ORDER DIFFERENTIAL EQUATIONS IN THE PHASE PLANE 1 1.1. Phase diagram for the pendulum equation 1 1.2. Autonomous equations in the phase plane 5 1.3. Conservative systems 10 1.4. The damped linear oscillator 12 1.5. Nonlinear damping 16 1.6. Some applications 20 1.7. Parameter-dependent conservative systems 25 Exercises 29 2. FIRST-ORDER SYSTEMS IN TWO VARIABLES AND LINEARIZATION 36 2.1. The general phase plane 36 2.2. Some population models 39 2.3. Linear approximation at equilibrium points 43 2.4. The general solution of a linear system 44 2.5. Classifying equilibrium points 45 2.6. Constructing a phase diagram 51 2.7. Transitions between types of equilibrium point 53 Exercises 54 3. GEOMETRICAL AND COMPUTATIONAL ASPECTS OF THE PHASE DIAGRAM 65 3.1. The index of a point 65 3.2. The index at infinity 74 3.3. The phase diagram at infinity 76 3.4. Limit cycles and other closed paths 81 3.5. Computation of the phase diagram 84 Exercises 88 4. AVERAGING METHODS 98 4.1. An energy-balance method for limit cycles 98 4.2. Amplitude and frequency estimates 102 4.3. Slowly-varying amplitude: nearly periodic solutions 105 4.4. Periodic solutions: harmonic balance 109 4.5. The equivalent linear equation by harmonic balance 110 Exercises 113 5. PERTURBATION METHODS 120 5.1. Outline of the direct method 120
Contents viii 5.2. Forced oscillations far from resonance 122 5.3. Forced oscillations near resonance with weak excitation 123 5.4. The amplitude equation for the undamped pendulum 126 5.5. The amplitude equation for a damped pendulum 129 5.6. Soft and hard springs 130 5.7. Amplitude-phase perturbation for the pendulum equation 134 5.8. Periodic solutions of autonomous equations (Lindstedt’s method) 136 5.9. Forced oscillation of a self-excited equation 138 5.10. The perturbation method and Fourier series 140 Exercises 142 6. SINGULAR PERTURBATION METHODS 146 6.1. Non-uniform approximations to functions on an interval 146 6.2. Coordinate perturbation (renormalization) 148 6.3. Lighthill’s method 153 6.4. Multiple time scales (two-timing) 155 6.5. Matching approximations on an interval 160 6.6. A matching technique for differential equations 165 Exercises 168 7. FORCED OSCILLATIONS: HARMONIC AND SUBHARMONIC RESPONSE, STABILITY, ENTRAINMENT 174 7.1. General forced periodic solutions 174 7.2. Harmonic solutions, transients, and stability for Duffing’s equation 176 7.3. The jump phenomenon 183 7.4. Harmonic oscillations, stability, and transients for the forced van der Pol equation 186 7.5. Frequency entrainment for the van der Pol equation 191 7.6. Comparison of the theory with computations 194 7.7. Subharmonics of Duffing’s equation by perturbation 195 7.8. Stability and transients for subharmonics of Duffing’s equation 200 Exercises 205 8. STABILITY 212 8.1. Poincaré stability (stability of paths) 213 8.2. Paths and solution curves 218 8.3. Liapunov stability (stability of solutions) 221 8.4. Stability of linear systems 225
Contents ix 8.5. Structure of the solutions of «-dimensional linear systems 227 8.6. Stability and boundedness for linear systems 234 8.7. Stability of systems with constant coefficients 235 Exercises 239 9. DETERMINATION OF STABILITY BY SOLUTION PERTURBATION 243 9.1. The stability of forced oscillations by solution perturbation 243 9.2. Equations with periodic coefficients (Floquet theory) 245 9.3. Mathieu’s equation arising from a Duffing equation 251 9.4. Transition curves for Mathieu’s equation by perturbation 256 9.5. Mathieu’s damped equation arising from a Duffing equation 257 Exercises 261 10. LIAPUNOV METHODS FOR DETERMINING STABILITY 267 10.1. Liapunov’s direct method 267 10.2. Liapunov functions 268 10.3. A test for instability 272 10.4. Stability and the linear approximation in two dimensions 273 10.5. Special systems 281 Exercises 285 11. THE EXISTENCE OF PERIODIC SOLUTIONS 293 11.1. The Poincaré-Bendixson theorem 293 11.2. A theorem on the existence of a centre 301 11.3. A theorem on the existence of a limit cycle , 305 11.4. Van der Pol’s equation with large parameter 311 Exercises 314 12. BIFURCATIONS, STRUCTURAL STABILITY, AND CHAOS 317 12.1. Examples of bifurcations 317 12.2. The fold and the cusp 319 12.3. Structural stability and bifurcations 323 12.4. Hopf bifurcations 327 12.5. Poincare maps 329 12.6. Chaos and strange attractors 336 12.7. Perturbation analysis of an amplitude bifurcation 345 12.8. Homoclinic bifurcation 347 Exercises 357 APPENDIX A: Existence and uniqueness theorems 362 APPENDIX B: Hints and answers to the exercises 365 BIBLIOGRAPHY 373 INDEX 377
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| any_adam_object | 1 |
| author | Jordan, Dominic W. Smith, Peter 1935- |
| author_GND | (DE-588)115172602 (DE-588)141762349 |
| author_facet | Jordan, Dominic W. Smith, Peter 1935- |
| author_role | aut aut |
| author_sort | Jordan, Dominic W. |
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| bvnumber | BV002284917 |
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| ctrlnum | (OCoLC)13904195 (DE-599)BVBBV002284917 |
| dewey-full | 515.3/52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.3/52 |
| dewey-search | 515.3/52 |
| dewey-sort | 3515.3 252 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| indexdate | 2024-07-09T15:43:22Z |
| institution | BVB |
| isbn | 019859657X 0198596561 |
| language | English |
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| physical | IX, 381 S. graph. Darst. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
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| series2 | Oxford applied mathematics and computing science series |
| spelling | Jordan, Dominic W. Verfasser (DE-588)115172602 aut Nonlinear ordinary differential equations D. W. Jordan and P. Smith 2. ed. Oxford Clarendon Press 1987 IX, 381 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford applied mathematics and computing science series Equations différentielles non linéaires ram Équations différentielles non linéaires Differential equations, Nonlinear Nichtlineare gewöhnliche Differentialgleichung (DE-588)4478411-9 gnd rswk-swf Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Nichtlineare gewöhnliche Differentialgleichung (DE-588)4478411-9 s DE-604 Nichtlineare Differentialgleichung (DE-588)4205536-2 s Gewöhnliche Differentialgleichung (DE-588)4020929-5 s 1\p DE-604 Smith, Peter 1935- Verfasser (DE-588)141762349 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001501655&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Jordan, Dominic W. Smith, Peter 1935- Nonlinear ordinary differential equations Equations différentielles non linéaires ram Équations différentielles non linéaires Differential equations, Nonlinear Nichtlineare gewöhnliche Differentialgleichung (DE-588)4478411-9 gnd Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4478411-9 (DE-588)4205536-2 (DE-588)4020929-5 |
| title | Nonlinear ordinary differential equations |
| title_auth | Nonlinear ordinary differential equations |
| title_exact_search | Nonlinear ordinary differential equations |
| title_full | Nonlinear ordinary differential equations D. W. Jordan and P. Smith |
| title_fullStr | Nonlinear ordinary differential equations D. W. Jordan and P. Smith |
| title_full_unstemmed | Nonlinear ordinary differential equations D. W. Jordan and P. Smith |
| title_short | Nonlinear ordinary differential equations |
| title_sort | nonlinear ordinary differential equations |
| topic | Equations différentielles non linéaires ram Équations différentielles non linéaires Differential equations, Nonlinear Nichtlineare gewöhnliche Differentialgleichung (DE-588)4478411-9 gnd Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Equations différentielles non linéaires Équations différentielles non linéaires Differential equations, Nonlinear Nichtlineare gewöhnliche Differentialgleichung Nichtlineare Differentialgleichung Gewöhnliche Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001501655&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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