Nonlinear Dynamics and Chaos

Description

The goal of this third edition of Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineeringis the same as previous editions: to provide a good foundation - and a joyful experience - for anyone who’d like to learn about nonlinear dynamics and chaos from an applied perspective. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. The prerequisites are comfort with multivariable calculus and linear algebra, as well as a first course in physics. Ideas from probability, complex analysis, and Fourier analysis are invoked, but they're either worked out from scratch or can be safely skipped (or accepted on faith). Changes to this edition include substantial exercises about conceptual models of climate change, an updated treatment of the SIR model of epidemics, and amendments (based on recent research) about the Selkov model of oscillatory glycolysis. Equations, diagrams, and every word has been reconsidered and often revised. There are also about 50 new references, many of them from the recent literature. The most notable change is a new chapter. Chapter 13 is about the Kuramoto model. The Kuramoto model is an icon of nonlinear dynamics. Introduced in 1975 by the Japanese physicist Yoshiki Kuramoto, his elegant model is one of the rare examples of a high-dimensional nonlinear system that can be solved by elementary means. Students and teachers have embraced the book in the past, its general approach and framework continue to be sound. Cover Half Title Title Page Copyright Page Contents Preface to the Third Edition Preface to the Second Edition Preface to the First Edition 1. Overview 1.0. Chaos, Fractals, and Dynamics 1.1. Capsule History of Dynamics 1.2. The Importance of Being Nonlinear 1.3. A Dynamical View of the World Part I: One-Dimensional Flows 2. Flows on the Line 2.0. Introduction 2.1. A Geometric Way of Thinking 2.2. Fixed Points and Stability 2.3. Population Growth 2.4. Linear Stability Analysis 2.5. Existence and Uniqueness 2.6. Impossibility of Oscillations 2.7. Potentials 2.8. Solving Equations on the Computer 3. Bifurcations 3.0. Introduction 3.1. Saddle-Node Bifurcation 3.2. Transcritical Bifurcation 3.3. Laser Threshold 3.4. Pitchfork Bifurcation 3.5. Overdamped Bead on a Rotating Hoop 3.6. Imperfect Bifurcations and Catastrophes 3.7. Insect Outbreak 4. Flows on the Circle 4.0. Introduction 4.1. Examples and Definitions 4.2. Uniform Oscillator 4.3. Nonuniform Oscillator 4.4. Overdamped Pendulum 4.5. Fireflies 4.6. Superconducting Josephson Junctions Part II: Two-Dimensional Flows 5. Linear Systems 5.0. Introduction 5.1. Definitions and Examples 5.2. Classification of Linear Systems 5.3. Love Affairs 6. Phase Plane 6.0. Introduction 6.1. Phase Portraits 6.2. Existence, Uniqueness, and Topological Consequences 6.3. Fixed Points and Linearization 6.4. Rabbits versus Sheep 6.5. Conservative Systems 6.6. Reversible Systems 6.7. Pendulum 6.8. Index Theory 7. Limit Cycles 7.0. Introduction 7.1. Examples 7.2. Ruling Out Closed Orbits 7.3. Poincaré–Bendixson Theorem 7.4. Liénard Systems 7.5. Relaxation Oscillations 7.6. Weakly Nonlinear Oscillators 8. Bifurcations Revisited 8.0. Introduction 8.1. Saddle-Node, Transcritical, and Pitchfork Bifurcations 8.2. Hopf Bifurcations 8.3. Oscillating Chemical Reactions 8.4. Global Bifurcations of Cycles 8.5. Hysteresis in the Driven Pendulum and Josephson Junction 8.6. Coupled Oscillators and Quasiperiodicity 8.7. Poincaré Maps Part III: Chaos 9. Lorenz Equations 9.0. Introduction 9.1. A Chaotic Waterwheel 9.2. Simple Properties of the Lorenz Equations 9.3. Chaos on a Strange Attractor 9.4. Lorenz Map 9.5. Exploring Parameter Space 9.6. Using Chaos to Send Secret Messages 10. One-Dimensional Maps 10.0. Introduction 10.1. Fixed Points and Cobwebs 10.2. Logistic Map: Numerics 10.3. Logistic Map: Analysis 10.4. Periodic Windows 10.5. Liapunov Exponent 10.6. Universality and Experiments 10.7. Renormalization 11. Fractals 11.0. Introduction 11.1. Countable and Uncountable Sets 11.2. Cantor Set 11.3. Dimension of Self-Similar Fractals 11.4. Box Dimension 11.5. Pointwise and Correlation Dimensions 12. Strange Attractors 12.0. Introduction 12.1. The Simplest Examples 12.2. Hénon Map 12.3. Rössler System 12.4. Chemical Chaos and Attractor Reconstruction 12.5. Forced Double-Well Oscillator Part IV: Collective Behavior 13. Kuramoto Model 13.0. Introduction 13.1. Governing Equations 13.2. Visualization and the Order Parameter 13.3. Mean-Field Coupling and Rotating Frame 13.4. Steady State 13.5. Self-Consistency 13.6. Remaining Questions Answers to Selected Exercises References Author Index Subject Index