Introduction to Nonlinear Physics

This textbook provides an introduction to the new science of nonlinear physics for advanced undergraduates, beginning graduate students, and researchers entering the field. The chapters, by pioneers and experts in the field, share a unified perspective. Nonlinear science developed out of the increasing ability to investigate and analyze systems for which effects are not simply linear functions of their causes; it is associated with such well-known code words as chaos, fractals, pattern formation, solitons, cellular automata, and complex systems. Nonlinear phenomena are important in many fields, including dynamical systems, fluid dynamics, materials science, statistical physics, and paritcel physics. The general principles developed in this text are applicable in a wide variety of fields in the natural and social sciences. The book will thus be of interest not only to physicists, but also to engineers, chemists, geologists, biologists, economists, and others interested in nonlinear phenomena. Examples and exercises complement the text, and extensive references provide a guide to research in the field.

Preface.- 1.1 A Quiet Revolution.- 1.2 Nonlinearity.- 1.3 Nonlinear Science.- 1.3.1 Fractals.- 1.3.2 Chaos.- 1.3.3 Pattern Formation.- 1.3.4 Solitons.- 1.3.5 Cellular Automata.- 1.3.6 Complex Systems.- 1.4 Remarks.- References.- Fractals and Multifractals.- Fractals and Diffusive Growth.- 2.1 Percolation.- 2.2 Diffusion-Limited Aggregation.- 2.3 Electrostatic Analogy.- 2.4 Physical Applications of DLA.- 2.4.1 Electrodeposition with Secondary Current Distribution.- 2.4.2 Diffusive Electrodeposition.- Problems.- References.- Multifractality.- 3.1 Definition of i(#)and/(a).- 3.2 Systematic Definition of x(q).- 3.3 The Two-Scale Cantor Set.- 3.3.1 Limiting Cases.- 3.3.2 Stirling Formula and/(a).- 3.4 Multifractal Correlations.- 3.4.1 Operator Product Expansion and Multifractality.- 3.4.2 Correlations of Iso-a Sets.- 3.5 Numerical Measurements of/(a).- 3.6 Ensemble Averaging and r(q).- Problems.- References.- Scaling Arguments and Diffusive Growth.- 4.1 The Information Dimension.- 4.2 The Turkevich-Scher Scaling Relation.- 4.3 The Electrostatic Scaling Relation.- 4.4 Scaling of Negative Moments.- 4.5 Conclusions.- Problems.- References.- Chaos and Randomness.- to Dynamical Systems.- 5.1 Introduction.- 5.2 Determinism Versus Random Processes.- 5.3 Scope of Part II.- 5.4 Deterministic Dynamical Systems and State Space.- 5.5 Classification.- 5.5.1 Properties of Dynamical Systems.- 5.5.2 A Brief Taxonomy of Dynamical Systems Models.- 5.5.3 The Relationship Between Maps and Flows.- 5.6 Dissipative Versus Conservative Dynamical Systems.- 5.7 Stability.- 5.7.1 Linearization.- 5.7.2 The Spectrum of Lyapunov Exponents.- 5.7.3 Invariant Sets.- 5.7.4 Attractors.- 5.7.5 Regular Attractors.- 5.7.6 Review of Stability.- 5.8 Bifurcations.- 5.9 Chaos.- 5.9.1 Binary Shift Map.- 5.9.2 Chaos in Flows.- 5.9.3 The Rossler Attractor.- 5.9.4 The Lorenz Attractor.- 5.9.5 Stable and Unstable Manifolds.- 5.10 Homoclinic Tangle.- 5.10.1 Chaos in Higher Dimensions.- 5.10.2 Bifurcations Between Chaotic Attractors.- Problems.- References.- Probability, Random Processes, and the Statistical Description of Dynamics.- 6.1 Nondeterminism in Dynamics.- 6.2 Measure and Probability.- 6.2.1 Estimating a Density Function from Data.- 6.3 Nondeterministic Dynamics.- 6.4 Averaging.- 6.4.1 Stationarity.- 6.4.2 Time Averages and Ensemble Averages.- 6.4.3 Mixing.- 6.5 Characterization of Distributions.- 6.5.1 Moments.- 6.5.2 Entropy and Information.- 6.6 Fractals, Dimension, and the Uncertainty Exponent.- 6.6.1 Pointwise Dimension.- 6.6.2 Information Dimension.- 6.6.3 Fractal Dimension.- 6.6.4 Generalized Dimensions.- 6.6.5 Estimating Dimension from Data.- 6.6.6 Embedding Dimension.- 6.6.7 Fat Fractals.- 6.6.8 Lyapunov Dimension.- 6.6.9 Metric Entropy.- 6.6.10 Pesin’s Identity.- 6.7 Dimensions, Lyapunov Exponents, and Metric Entropy in the Presence of Noise.- Problems.- References.- Modeling Chaotic Systems.- 7.1 Chaos and Prediction.- 7.2 State Space Reconstruction.- 7.2.1 Derivative Coordinates.- 7.2.2 Delay Coordinates.- 7.2.3 Broomhead and King Coordinates.- 7.2.4 Reconstruction as Optimal Encoding.- 7.3 Modeling Chaotic Dynamics.- 7.3.1 Choosing an Appropriate Model.- 7.3.2 Order of Approximation.- 7.3.3 Scaling of Errors.- 7.4 System Characterization.- 7.5 Noise Reduction.- 7.5.1 Shadowing.- 7.5.2 Optimal Solution of Shadowing Problem with Euclidean Norm.- 7.5.3 Numerical Results.- 7.5.4 Statistical Noise Reduction.- 7.5.5 Limits to Noise Reduction.- Problems.- References.- Pattern Formation and Disorderly Growth.- Phenomenology of Growth.- 8.1 Aggregation: Patterns and Fractals Far from Equilibrium.- 8.2 Natural Systems.- 8.2.1 Ballistic Growth.- 8.2.2 Diffusion-Limited Growth.- 8.2.3 Growth of Colloids and Aerosols.- Problems.- References.- Models and Applications.- 9.1 Ballistic Growth.- 9.1.1 Simulations and Scaling.- 9.1.2 Continuum Models.- 9.2 Diffusion-Limited Growth.- 9.2.1 Simulations and Scaling.- 9.2.2 The Mullins-Sekerka Instability.- 9.2.3 Orderly and Disorderly Growth.- 9.2.4 Electrochemical Deposition: A Case Study.- 9.3 Cluster-Cluster Aggregation.- Appendix: A DLA Program.- Problems.- References.- Solitons.- Models and Applications.- 10.1 Introduction.- 10.2 Origin and History of Solitons.- 10.3 Integrability and Conservation Laws.- 10.4 Soliton Equations and their Solutions.- 10.4.1 Korteweg-de Vries Equation.- 10.4.2 Nonlinear Schrodinger Equation.- 10.4.3 Sine-Gordon Equation.- 10.4.4 Kadomtsev-Petviashvili Equation.- 10.5 Methods of Solution.- 10.5.1 Inverse Scattering Method.- 10.5.2 Backlund Transformation.- 10.5.3 Hirota Method.- 10.5.4 Numerical Method.- 10.6 Physical Soliton Systems.- 10.6.1 Shallow Water Waves.- 10.6.2 Dislocations in Crystals.- 10.6.3 Self-Focusing of Light.- 10.7 Conclusions.- Problems.- References.- Nonintegrable Systems.- 11.1 Introduction.- 11.2 Nonintegrable Soliton Equations with Hamiltonian Structures.- 11.2.1 The fl4 Equation.- 11.2.2 Double Sine-Gordon Equation.- 11.3 Nonlinear Evolution Equations.- 11.3.1 Fisher Equation.- 11.3.2 The Damped 0* Equation.- 11.3.3 The Damped Driven Sine-Gordon Equation.- 11.4 A Method of Constructing Soliton Equations.- 11.5 Formation of Solitons.- 11.6 Perturbations.- 11.7 Soliton Statistical Mechanics.- 11.7.1 The ^System.- 11.7.2 The Sine-Gordon System.- 11.8 Solitons in Condensed Matter.- 11.8.1 Liquid Crystals.- 11.8.2 Polyacetylene.- 11.8.3 Optical Fibers.- 11.8.4 Magnetic Systems.- 11.9 Conclusions.- Problems.- References.- Special Topics.- Cellular Automata and Discrete Physics.- 12.1 Introduction.- 12.1.1 A Weil-Known Example: Life.- 12.1.2 Cellular Automata.- 12.1.3 The Information Mechanics Group.- 12.2 Physical Modeling.- 12.2.1 CA Quasiparticles.- 12.2.2 Physical Properties from CA Simulations.- 12.2.3 Diffusion.- 12.2.4 Soundwaves.- 12.2.5 Optics.- 12.2.6 Chemical Reactions.- 12.3 Hardware.- 12.4 Current Sources of Literature.- 12.5 An Outstanding Problem in CA Simulations.- Problems.- References.- Visualization Techniques for Cellular Dynamata.- 13.1 Historical Introduction.- 13.2 Cellular Dynamata.- 13.2.1 Dynamical Schemes.- 13.2.2 Complex Dynamical Systems.- 13.2.3 CD Definitions.- 13.2.4 CD States.- 13.2.5 CD Simulation.- 13.2.6 CD Visualization.- 13.3 An Example of Zeeman’s Method.- 13.3.1 Zeeman’s Heart Model: Standard Cell.- 13.3.2 Zeeman’s Heart Model: Physical Space.- 13.3.3 Zeeman’s Heart Model: Beating.- 13.4 The Graph Method.- 13.4.1 The Biased Logistic Scheme.- 13.4.2 The Logistic/Diffusion Lattice.- 13.4.3 The Global State Graph.- 13.5 The Isochron Coloring Method.- 13.5.1 Isochrons of a Periodic Attractor.- 13.5.2 Coloring Strategies.- 13.6 Conclusions.- References.- From Laminar Flow to Turbulence.- 14.1 Preamble and Basic Ideas.- 14.1.1 What Is Turbulence?.- 14.2 From Laminar Flow to Nonlinear Equilibration.- 14.2.1 A Linear Analysis: The Kelvin-Helmholz Instability.- 14.2.2 A Weakly Nonlinear Analysis: Landau’s Equation.- 14.3 From Nonlinear Equilibration to Weak Turbulence.- 14.3.1 The Quasi-Periodic Sequence.- 14.3.2 The Period Doubling Sequence.- 14.3.3 The Intermittent Sequence.- 14.3.4 Fluid Relevance and Experimental Evidence.- 14.4 Strong Turbulence.- 14.4.1 Scaling Arguments for Inertial Ranges.- 14.4.2 Predictability of Strong Turbulence.- 14.4.3 Renormalizing the Diffusivity.- 14.5 Remarks.- References.- Active Walks: Pattern Formation, Self-Organization, and Complex Systems.- 15.1 Introduction.- 15.2 Basic Concepts.- 15.3 Continuum Description.- 15.4 Computer Models.- 15.4.1 A Single Walker.- 15.4.2 Branching.- 15.4.3 Multiwalkers and Updating Rules.- 15.4.4 Track Patterns.- 15.5 Three Applications.- 15.5.1 Dielectric Breakdown in a Thin Layer of Liquid.- 15.5.2 Ion Transport in Glasses.- 15.5.3 Ant Trails in Food Collection.- 15.6 Intrinsic Abnormal Growth.- 15.7 Landscapes and Rough Surfaces.- 15.7.1 Groove States.- 15.7.2 Localization-Delocalization Transition.- 15.7.3 Scaling Properties.- 15.8 Fuzzy Walks.- 15.9 Related Developments and Open Problems.- 15.10 Conclusions.- References.- Appendix: Historical Remarks on Chaos.- Contributors.