Nonlinear differential equations and dynamical systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Dutch |
| Published: |
Berlin [u.a.]
Springer
1996
|
| Edition: | 2., rev. and expanded ed. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 295 - 300 |
| Physical Description: | X, 303 S. graph. Darst. |
| ISBN: | 3540609342 |
Staff View
MARC
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| 100 | 1 | |a Verhulst, Ferdinand |d 1939- |e Verfasser |0 (DE-588)111117755 |4 aut | |
| 240 | 1 | 0 | |a Nietlineaire differentiaalvergelijkingen en dynamische systemen |
| 245 | 1 | 0 | |a Nonlinear differential equations and dynamical systems |c Ferdinand Verhulst |
| 250 | |a 2., rev. and expanded ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1996 | |
| 300 | |a X, 303 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Universitext | |
| 500 | |a Literaturverz. S. 295 - 300 | ||
| 650 | 7 | |a Differentiaalvergelijkingen |2 gtt | |
| 650 | 7 | |a Dynamique différentielle |2 ram | |
| 650 | 7 | |a Dynamische systemen |2 gtt | |
| 650 | 7 | |a Equations différentielles non linéaires |2 ram | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Introduction 1
1.1 Definitions and notation 1
1.2 Existence and uniqueness 3
1.3 Gronwall s inequality 4
2 Autonomous equations 7
2.1 Phase space, orbits 7
2.2 Critical points and linearisation 10
2.3 Periodic solutions 14
2.4 First integrals and integral manifolds 16
2.5 Evolution of a volume element, Liouville s theorem 21
2.6 Exercises 23
3 Critical points 25
3.1 Two dimensional linear systems 25
3.2 Remarks on three dimensional linear systems 29
3.3 Critical points of nonlinear equations 31
3.4 Exercises 36
4 Periodic solutions 38
4.1 Bendixson s criterion 38
4.2 Geometric auxiliaries, preparation for the
Poincare Bendixson theorem 40
4.3 The Poincare Bendixson theorem 43
4.4 Applications of the Poincare Bendixson theorem 47
4.5 Periodic solutions in R 53
4.6 Exercises 57
5 Introduction to the theory of stability 59
5.1 Simple examples 59
5.2 Stability of equilibrium solutions 61
5.3 Stability of periodic solutions 62
5.4 Linearisation 66
5.5 Exercises 67
6 Linear Equations 69
6.1 Equations with constant coefficients 69
6.2 Equations with coefficients which have a limit 71
VIII Contents
6.3 Equations with periodic coefficients 75
6.4 Exercises 80
7 Stability by linearisation 83
7.1 Asymptotic stability of the trivial solution 83
7.2 Instability of the trivial solution 88
7.3 Stability of periodic solutions of autonomous equations 91
7.4 Exercises 93
8 Stability analysis by the direct method 96
8.1 Introduction 96
8.2 Lyapunov functions 98
8.3 Hamiltonian systems and systems with first integrals 103
8.4 Applications and examples 107
8.5 Exercises 108
9 Introduction to perturbation theory 110
9.1 Background and elementary examples 110
9.2 Basic material 113
9.3 Naive expansion 116
9.4 The Poincare expansion theorem 119
9.5 Exercises 120
10 The Poincare Lindstedt method 122
10.1 Periodic solutions of autonomous second order equations 122
10.2 Approximation of periodic solutions
on arbitrary long time scales 127
10.3 Periodic solutions of equations with forcing terms 129
10.4 The existence of periodic solutions 131
10.5 Exercises 135
11 The method of averaging 136
11.1 Introduction 136
11.2 The Lagrange standard form 138
11.3 Averaging in the periodic case 140
11.4 Averaging in the general case 144
11.5 Adiabatic invariants 147
11.6 Averaging over one angle, resonance manifolds 150
11.7 Averaging over more than one angle, an introduction 154
11.8 Periodic solutions 157
11.9 Exercises 162
Contents IX
12 Relaxation Oscillations 166
12.1 Introduction 166
12.2 Mechanical systems with large friction 167
12.3 The van der Pol equation 168
12.4 The Volterra Lotka equations 170
12.5 Exercises 172
13 Bifurcation Theory 173
13.1 Introduction 173
13.2 Normalisation 175
13.3 Averaging and normalisation 180
13.4 Centre manifolds 182
13.5 Bifurcation of equilibrium solutions
and Hopf bifurcation 186
13.6 Exercises 190
14 Chaos 193
14.1 Introduction and historical context 193
14.2 The Lorenz equations 194
14.3 Maps associated with the Lorenz equations 197
14.4 One dimensional dynamics 199
14.5 One dimensional chaos: the quadratic map 203
14.6 One dimensional chaos: the tent map 207
14.7 Fractal sets 208
14.8 Dynamical characterisations of fractal sets 213
14.9 Lyapunov exponents 216
14.10Ideas and references to the literature 218
15 Hamiltonian systems 224
15.1 Introduction 224
15.2 A nonlinear example with two degrees of freedom 226
15.3 Birkhoff normalisation 230
15.4 The phenomenon of recurrence 233
15.5 Periodic solutions 236
15.6 Invariant tori and chaos 238
15.7 The KAM theorem 242
15.8 Exercises 246
Appendix 1: The Morse lemma 248
Appendix 2: Linear periodic equations with a small parameter 250
Appendix 3: Trigonometric formulas and averages 252
X Contents
Appendix 4: A sketch of Cotton s proof of the stable
and unstable manifold theorem 3.3 253
Appendix 5: Bifurcations of self excited oscillations 255
Appendix 6: Normal forms of Hamiltonian systems
near equilibria 260
Answers and hints to the exercises 267
References 295
Index 301
|
| any_adam_object | 1 |
| author | Verhulst, Ferdinand 1939- |
| author_GND | (DE-588)111117755 |
| author_facet | Verhulst, Ferdinand 1939- |
| author_role | aut |
| author_sort | Verhulst, Ferdinand 1939- |
| author_variant | f v fv |
| building | Verbundindex |
| bvnumber | BV010903751 |
| callnumber-first | Q - Science |
| callnumber-label | QA372 |
| callnumber-raw | QA372.V4713 1996 |
| callnumber-search | QA372.V4713 1996 |
| callnumber-sort | QA 3372 V4713 41996 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 500 SK 520 |
| classification_tum | MAT 344f MAT 340f PHY 013f MAT 354f |
| ctrlnum | (OCoLC)34617831 (DE-599)BVBBV010903751 |
| dewey-full | 515/.35520 515/.355 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.355 20 515/.355 |
| dewey-search | 515/.355 20 515/.355 |
| dewey-sort | 3515 3355 220 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | 2., rev. and expanded ed. |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:00:53Z |
| institution | BVB |
| isbn | 3540609342 |
| language | English Dutch |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007293028 |
| oclc_num | 34617831 |
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| physical | X, 303 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Springer |
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| series2 | Universitext |
| spelling | Verhulst, Ferdinand 1939- Verfasser (DE-588)111117755 aut Nietlineaire differentiaalvergelijkingen en dynamische systemen Nonlinear differential equations and dynamical systems Ferdinand Verhulst 2., rev. and expanded ed. Berlin [u.a.] Springer 1996 X, 303 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Literaturverz. S. 295 - 300 Differentiaalvergelijkingen gtt Dynamique différentielle ram Dynamische systemen gtt Equations différentielles non linéaires ram Equações diferenciais não lineares larpcal Niet-lineaire vergelijkingen gtt Differential equations, Nonlinear Differentiable dynamical systems Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Nichtlineare Differentialgleichung (DE-588)4205536-2 s DE-604 Dynamisches System (DE-588)4013396-5 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007293028&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Verhulst, Ferdinand 1939- Nonlinear differential equations and dynamical systems Differentiaalvergelijkingen gtt Dynamique différentielle ram Dynamische systemen gtt Equations différentielles non linéaires ram Equações diferenciais não lineares larpcal Niet-lineaire vergelijkingen gtt Differential equations, Nonlinear Differentiable dynamical systems Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4205536-2 (DE-588)4013396-5 |
| title | Nonlinear differential equations and dynamical systems |
| title_alt | Nietlineaire differentiaalvergelijkingen en dynamische systemen |
| title_auth | Nonlinear differential equations and dynamical systems |
| title_exact_search | Nonlinear differential equations and dynamical systems |
| title_full | Nonlinear differential equations and dynamical systems Ferdinand Verhulst |
| title_fullStr | Nonlinear differential equations and dynamical systems Ferdinand Verhulst |
| title_full_unstemmed | Nonlinear differential equations and dynamical systems Ferdinand Verhulst |
| title_short | Nonlinear differential equations and dynamical systems |
| title_sort | nonlinear differential equations and dynamical systems |
| topic | Differentiaalvergelijkingen gtt Dynamique différentielle ram Dynamische systemen gtt Equations différentielles non linéaires ram Equações diferenciais não lineares larpcal Niet-lineaire vergelijkingen gtt Differential equations, Nonlinear Differentiable dynamical systems Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Differentiaalvergelijkingen Dynamique différentielle Dynamische systemen Equations différentielles non linéaires Equações diferenciais não lineares Niet-lineaire vergelijkingen Differential equations, Nonlinear Differentiable dynamical systems Nichtlineare Differentialgleichung Dynamisches System |
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