.

0 (hodnocen0 x )

(2) Půjčeno:2x 

BK

Berlin : Springer, 1996

x, 303 s.

ISBN 3-540-60934-2

000069718

1 Introduction 1 // 1.1 Definitions and notation 1 // 1.2 Existence and uniqueness 3 // 1.3 Gronwalls 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 Mn 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 // vin // 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 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 // 14.10Ideas 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