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Nonlinear Differential Equations and Dynamical Systems

Edition: 
2
Publisher: 
Springer Verlag
Number of Pages: 
303
Price: 
39.95
ISBN: 
3540609342
Date Received: 
Thursday, April 27, 2006
Reviewable: 
Yes
Include In BLL Rating: 
Yes
Ferdinand Verhulst
Series: 
Universitext
Publication Date: 
2006
Format: 
Paperback
Category: 
Monograph
BLL
10/19/2010
BLL Rating: 

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 
11.5 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 
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
Publish Book: 
Modify Date: 
Tuesday, October 19, 2010

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