Introduction to Nonlinear Science

Cover......Page 1
Introduction to nonlinear science......Page 4
Contents......Page 7
Preface......Page 10
I . I What is nonlinearity?......Page 13
1.2 Nonlinear behavior in classical mechanics......Page 14
1.3 Thermal convection......Page 17
1.4 Nonlinear phenomena in chemistry......Page 24
1.5 Some further examples of chemically mediated nonlinear behavior.......Page 31
Problems......Page 35
2.1 Evolution equations in classical mechanics......Page 37
2.2 The macroscopic level: balance equation of a macrovariable......Page 41
2.3 Conserved variables in a one-component system and the equations of fluid dynamics......Page 42
2.4 Nonconserved variables in a multicomponent system and the equations of chemical kinetics......Page 45
2.5 The Benard problem: quantitative formulation......Page 47
2.6 Some representative chemical models giving rise to nonlinear behavior......Page 52
Problems......Page 57
3.1 General orientation......Page 59
3.2 Phase space......Page 61
3.3 Invariant manifolds......Page 63
3.4 Conservative and dissipative systems. Attractors......Page 70
3.5 Stabi l i ty......Page 73
3.6 The principle of linearized stability......Page 78
Problems......Page 81
4.1 General formulation......Page 83
4.2 Systems involving one variable......Page 87
4.3 Systems involving two variables......Page 89
4.4 Examples of stability analysis of two-dimensional dynamical systems......Page 96
4.5 Three variables and beyond......Page 99
Problems......Page 104
5.1 Introduction......Page 106
5.2 Expansion of the solutions in perturbation series: the case of zero eigenvalue......Page 108
5.3 The amplitude equation: transcritical bifurcation......Page 110
5.4 The amplitude equation: pitchfork bifurcation......Page 114
5.5 Limit point bifurcation......Page 116
5.6 Kinetic potential, sensitivity, structural stability......Page 117
5.7 The Hopf bifurcation......Page 122
5.8 Cascading bifurcations......Page 126
5.9 Normal forms and resonances......Page 132
Problems......Page 137
6.1 General formulation......Page 140
6.2 The Benard problem: reference state and linearization of the Boussinesq equations......Page 141
6.3 The Benard problem: linear stability analysis for free boundaries......Page 145
6.4 Reaction-diffusion systems. The Turing instability......Page 150
6.5 Further comments on linear stability in spatially distributed systems......Page 158
6.6 Bifurcation analysis: general formulation......Page 160
6.7 Bifurcation of two-dimensional rolls in the Benard problem: the small aspect ratio case......Page 163
6.8 Bifurcation analysis in systems of large spatial extent: complex Landau-Ginzburg equation......Page 168
6.9 Further examples of normal form envelope equations in large systems......Page 173
Problems......Page 181
7.1 The Poincare map......Page 185
7-2 One-dirnen~ionr~elc urrences: general aspects......Page 190
7.3 Phenomenology of one-dimensional recurrences: illustrations......Page 192
7.4 Tools of chaos theory......Page 200
7.5 Routes to chaos: quantitative formulation......Page 204
7.6 Fully developed chaos: probabilistic description......Page 208
7.7 Error growth, Lyapunov exponents and predictability......Page 217
7.8 The dynamics of symbolic sequences: entropy, master equation......Page 223
7.9 Spatio-temporal chaos......Page 232
Problems......Page 239
APPENDIX A1 Proof of theprinciple of linearized stability for one-variable systems......Page 242
APPENDIX A2 Hopf bifurcation analysis of the Brusselator model......Page 246
References......Page 251