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
Part I Fractals 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
Part II Chaos 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
5.7.6 ReviewofStability
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
6 Probability, Random Processes, and the
Statistfcal Description ofDynanucs
Stephen G. EubankandJ. Doyne Farmer
6.1 Nondeterminism in Dynamics
6.2 Measure and Probability
6.2.1 Estimating a Density Function from Data
6.3 Nondetenninistic Dynamics
6.4 Averaging
6.4.1 Stationarity
6.4.2 Time Averages and Ensemble Averages
6.4.3 Mixing
6.5 Characterization ofDistributions
6.5.1 Moments
6.5.2 Entropy and Infonnation
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 ofNoise
Problems
References
7 Modeling Chaotic Systems
Stephen G. Eubank and J. Doyne Farmer
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 OrderofApproximation
7.3.3 ScalingofErrors
7.4 System Characterization
7.5 Noise Reduction
7.5.1 Shadowing
7.5.2 Optimal Solution ofShadowing Problem
with Euclidean Nonn
7.5.3 Numerical Results
7.5.4 Statistical Noise Reduction
7.5.5 Limits to Noise Reduction
Problems
References
Part III Pattero Formation and Disorderly Growth
8 Phenomenology of Growth
Leonard M. Sander
8.1 Aggregation: Pattems and Fractals Far from Equilibrium
8.2 Natural Systems
8.2.1 Ballistic Growth
8.2.2 Diffusion-Limited Growth
8.2.3 GrowthofColloidsandAerosols
Problems
References
9 Models and Applications
Leonard M. Sander
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 Orderiy and Disorderiy Growth
9.2.4 Electrochemical Deposition: A Case Study
9.3 Cluster-Cluster Aggregation
Appendix: A DLA Program
Problems
References
Part IV SoBtons
10 Integrable Systems
LuiLam
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 Smc-Gordon Equation
10.4.4 Kadomtsev-Petviashvili Equation
10.5 MethodsofSolution
10.5.1 Inverse Scattering Method
10.5.2 Bficklund Transformation
10.5.3 Hirota Method
10.5.4 Numerical Method
10.6 Physical Soliton Systems
10.6.1 ShallowWaterWaves
10.6.2 Dislocations in Crystals
10.6.3 Self-FocusingofLight
10.7 Conclusions
Problems
References
11 Nonintegrable Systems
LuiLam
11.1 Introduction
11.2 Nonintegrable Soliton Equations with Hamiltonian Structures
11.2.1 The Equation
11.2.2 Double Sine-Gordon Equation
11.3 Nonlinear Evolution Equations
11.3.1 Fisher Equation
11.3.2 The Damped Equation
11.3.3 The Damped Driven Sine-Gordon Equation
11.4 A Method of Constructing Soliton Equations
11.5 FonnationofSolitons
11.6 Perturbations
11.7 Soliton Statistical Mechanics
11.7.1 TheSystem
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
Part V Special Topics
12 Cellular Automata and Discrete Physics
David E. Hiebeler and Robert Tatar
12.1 Introduction
12.1.1 A Well-Kaown 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
13 Visualization Techniques for Cellular Dynamata
Ralph H. Abraham
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 ofZeeman'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 ofa Periodic Attractor
13.5.2 Coloring Strategies
13.6 Conclusions
References
14 From Laminar Flow to Turbulence
GeoffreyK. Vallis
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
15 Active Walks: Pattern Formation, Self-Organization, and
Complex Systems LuiLam
15.1 Introduction
15.2 Basic Concepts
15.3 Continuum Description
15.4 Computer Models
15.4.1 ASingleWalker
15.4.2 Branching
15.4.3 Multiwalkers and Updating Rules
15.4.4 Track Pattems
15.5 Three Applications
15.5.1 Dielectric Breakdown in a Thin Layer ofLiquid
15.5.2 lon 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 GrooveStates
15.7.2 Localization-Delocalization Transition
15.7.3 Scaling Properties
15.8 FuzzyWalks
15.9 Related Developments and Open Problems
15.10 Conclusions
References
Appendix: Historical Remarks on Chaos
Michael Nauenberg
Contributors
Index
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