Introduction to Nonlinear Physics

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.