Contenidos

I. INTRODUCCIóN

• 1.0 Chaos, Fractals, and Dynamics 1
• 1.1 Capsule History of Dynamics 2
• 1.2 The Importance of Being Nonlinear 4
• 1.3 A Dynamical View of the World 9

Parte 1 - One-Dimensional Flows

II Flows on the Line 15 2.0 Introduction 15 2.1 A Geometric Way of Thinking 16 2.2 Fixed Points and Stability 18 2.3 Population Growth 21 2.4 Linear Stability Analysis 24 2.5 Existence and Uniqueness 26 2.6 Impossibility of Oscillations 28 2.7 Potentials 30 2.8 Solving Equations on the Computer Exercises for Chapter 2 36

III - Bifurcations 45 3.0 Introduction 45 3.1 Saddle-Node Bifurcation 46 3.2 Transcritical Bifurcation 51 3.3 Laser Threshold 54 3.4 Pitchfork Bifurcation 56 3.5 Overdamped Bead on a Rotating Hoop 62 3.6 Imperfect Bifurcations and Catastrophes 70 3.7 Insect Outbreak 74 Exercises for Chapter 3 80

IV Flows on the Circle 95 4.0 Introduction 95 4.1 Examples and Definitions 95 4.2 Uniform Oscillator 97 4.3 Nonuniform Oscillator 98 4.4 Overdamped Pendulum 103 4.5 Fireflies 105 4.6 Superconducting Josephson Junctions Exercises for Chapter 4 115

Parte 2 - Two-Dimensional Flows

V Linear Systems 125 5.0 Introduction 125 5.1 Definitions and Examples 125 5.2 Classification of Linear Systems 131 5.3 Love Affairs 139 Exercises for Chapter 5 142

VI Phase Plane 146 6.0 Introduction 146 6.1 Phase Portraits 146 6.2 Existence, Uniqueness, and Topological Consequences 149 6.3 Fixed Points and Linearization 151 6.4 Rabbits versus Sheep 156 6.5 Conservative Systems 160 6.6 Reversible Systems 164 6.7 Pendulum 168 6.8 Index Theory 174 Exercises for Chapter 6 181

VII Limit Cycles 198 7.0 Introduction 198 7.1 Examples 199 7.2 Ruling Out Closed Orbits 201 7.3 Poincaré−Bendixson Theorem 205 7.4 Liénard Systems 212 7.5 Relaxation Oscillations 213 7.6 Weakly Nonlinear Oscillators 217 Exercises for Chapter 7 230

VIII Bifurcations Revisited 244 8.0 Introduction 244 8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations 244 8.2 Hopf Bifurcations 251 8.3 Oscillating Chemical Reactions 257 8.4 Global Bifurcations of Cycles 264 8.5 Hysteresis in the Driven Pendulum and Josephson Junction 268 8.6 Coupled Oscillators and Quasiperiodicity 276 8.7 Poincaré Maps 281 Exercises for Chapter 8 287