Nonlinear dynamics: a two-way trip from physics to math

Description

Content: Cover Half Title Title Page Copyright Page Dedication Table of Contents Acknowledgments Preface 1: Nonlinear dynamics in nature 1.1 Hiking among rabbits 1.2 Turbulence 1.3 Bénard instability 1.4 Dynamics of a modulated laser 1.5 Tearing of a plasma sheet 1.6 Summary 2: Linear dynamics 2.1 Introduction 2.2 Why linear dynamics? 2.3 Linear flows 2.3.1 Autonomous flows 2.3.1.1 Autonomous homogeneous flows 2.3.1.2 Autonomous inhomogeneous flows 2.3.1.3 Singular points in bi-dimensional flows 2.3.2 Forced flows 2.3.2.1 Additively forced flows 2.3.2.2 Parametrically forced flows2.3.2.3 Floquet's theorem 2.3.2.4 Oscillation theorem-Liapunov-Haupt 2.3.2.5 Arnold tongues 2.4 Summary 2.5 Additional exercise 3: Nonlinear examples 3.1 Preliminary comments 3.2 A model for the CO2 laser 3.2.1 The model 3.2.2 Dynamics of the laser 3.3 Duffing oscillator 3.3.1 The model 3.3.2 Dynamics in the unforced model 3.4 The Lorenz equations 3.4.1 The model 3.4.1.1 Properties 3.4.2 Simple dynamics in the Lorenz model 3.5 Summary 3.6 Additional exercises 4: Elements of the description 4.1 Introduction 4.2 Basic elements 4.2.1 Phase space4.2.2 Flow 4.2.3 Invariants 4.2.3.1 Orbits 4.2.4 Attractors 4.2.4.1 Attracting sets 4.2.4.2 Attractors 4.2.4.3 Basin of attraction 4.2.4.4 Boundaries of the basin of attraction 4.2.5 Trapping regions 4.2.6 Stable and unstable sets 4.3 Poincaré sections 4.3.1 Stroboscopic section 4.3.2 Transverse section 4.3.3 Poincaré sections and Poincaré first-return maps 4.3.3.1 Global Poincaré section 4.3.3.2 Local Poincaré section 4.4 Maps and dynamics 4.4.1 Properties of maps 4.4.1.1 Conjugated maps 4.5 Parameter dependence 4.5.1 Families of flows and maps 4.6 Summary4.7 Additional exercise 5: Elementary stability theory 5.1 Introduction 5.2 Fixed point stability 5.2.1 Liapunov stability criteria 5.3 The validity of the linearization procedure 5.4 Maps and periodic orbits 5.4.1 Maps 5.4.2 Periodic orbits of flows, Floquet stability theory 5.5 Structural stability 5.5.1 Orbital equivalence 5.5.2 Structural equivalence 5.6 Summary 5.7 Additional exercise 6: Bi-dimensional flows 6.1 Limit sets 6.2 Transverse sections and sequences 6.3 Poincaré-Bendixson theorem 6.4 Structural stability 6.5 Summary 7: Bifurcations 7.1 The bifurcation programme7.1.1 Families of flows and maps 7.1.2 Local and global bifurcations 7.1.3 Local bifurcations: the programme 7.2 Equivalence between flows 7.3 Conditions for fixed point bifurcations 7.3.1 Unfolding of a bifurcation 7.4 Reduction to the centre manifold 7.4.1 Adiabatic elimination of fast-decaying variables 7.4.2 Approximation to the centre manifold 7.4.3 Centre manifold for maps 7.4.4 Bifurcations 7.5 Normal forms 7.5.1 Example: Jordan block 7.5.2 The general case 7.5.3 Normal forms for maps 7.5.4 Resonances 7.5.4.1 Small denominators