Nonlinear Differential Equations and Dynamical Systems

Ferdinand Verhulst
Publisher:
Springer Verlag
Publication Date:
2006
Number of Pages:
303
Format:
Paperback
Edition:
2
Series:
Universitext
Price:
39.95
ISBN:
3540609342
Category:
Monograph
BLL Rating:

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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
Poincaré-Bendixson theorem 40
4.3 The Poincaré-Bendixson theorem 43
4.4 Applications of the Poincaré-Bendixson theorem 47
4.5 Periodic solutions in R^n 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
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 Naïve expansion 116
9.4 The Poincaré expansion theorem 119
9.5 Exercises 120
10 The Poincaré-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
11.6 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
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
14.10 Ideas 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
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