Contents Preface 1 Introduction LuiLam 1.1 A Quiet Revolution 1.2 Nonlinearity 1.3 Nonlinear Science 1.3.1 Fractals 1.3.2 Chaos 1.3.3 Pattem Fonnation 1.3.4 Solitons 1.3.5 Cellular Automata 1.3.6 Complex Systems 1.4 Remarks References PartⅠFractals and Multifractals 2 Fractals and Diffusive Growth Thomas C. Halsey 2.1 Percolation 2.2 Diffusion-Limited Aggregation 2.3 Electrostatic Analogy 2.4 Physical Applications ofDLA 2.4.1 Electrodeposition with Secondary Current Distribution 2.4.2 Diffusive Electrodeposition Problems References 3 Multifractality Thomas C. Halsey 3.1 Defimtionof(q)and f(a) 3.2 SystematicDefinitionofT(q) 3.3 The Two-Scale Cantor Set 3.3.1 Limiting Cases 3.3.2 Stirling Formula andf(a) 3.4 Multifractal Correlations 3.4.1 Operator Product Expansion and Multifractality 3.4.2 Correlations oflso-flt Sets 3.5 Numerical Measurements of f(a) 3.6 Ensemble Averaging and (q) Problems References 4 Scaling Arguments and Diffusive Growth Thomas C. Halsey 4.1 The Information Dimension 4.2 The Turkevich-Scher Scaling Relation 4.3 The Electrostatic Scaling Relation 4.4 Scaling ofNegative Moments 4.5 Conclusions Problems References PartⅡChaos and Randomness 5 Introduction to Dynamical Systems Stephen G. Eubank and J. Doyne Farmer 5.1 Introduction 5.2 Detenninism Versus Random Processes 5.3 ScopeofPartⅡ 5.4 Deterministic Dynamical Systems and State Space 5.5 Classification 5.5.1 PropertiesofDynamical Systems 5.5.2 A BriefTaxonomy ofDynamical Systems Models 5.5.3 The Relationship Between Maps and Flows 5.6 Dissipative Versus Conservative Dynamical Systems 5.7 Stability 5.7.1 Lmearization 5.7.2 TheSpectrumofLyapunovExponents 5.7.3 InvariantSets 5.7.4 Attractors 5.7.5 Regular Attractors …… 6 Probability, Random Processes, and the 7 Modeling Chaotic Systems PartⅢPattero Formation and Disorderly Growth 8 Phenomenology of Growth 9 Models and Applications PartⅣSoBtons 10 Integrable Systems 11 Nonintegrable Systems PartⅤSpecial Topics 12 Cellular Automata and Discrete Physics 13 Visualization Techniques for Cellular Dynamata 14 From Laminar Flow to Turbulence 15 Active Walks: Pattern Formation, Self-Organization, and Appendix: Historical Remarks on Chaos Contributors Index
