Understanding nonlinear dynamics

SERIES PREFACE ABOUT THE AUTHORS PREFACE 1 FINITE-DIFFERENCE EQUATIONS 1.1 A Mythical Field 1.2 The Linear Finite-Difference Equation 1.3 Methods of Iteration 1.4 Nonlinear Finite-Difference Equations 1.5 Steady States and Their Stability 1.6 Cycles and Their Stability 1.7 Chaos 1.8 Quasiperiodicity 1 Chaos in Periodically Stimulated Heart Cells, Sources and Notes Exercises Computer Projects 2 BOOLEAN NETWORKS AND CELLULAR AUTOMATA 2.1 Elements and Networks 2.2 Boolean Variables, Functions, and Networks 2 A Lambda Bacteriophage Model 3 Locomotion in Salamander,a 2.3 Boolean Functions and Biochemistry 2.4 Random Boolean Networks 2.5 Cellular Automata 4 Spiral Waves in Chemistry and aiology 2.6 Advanced Topic: Evolution and Computation Sources and Notes Exercises Computer Projects 3 SELF-SIMILARITY AND FRACTAL GEOMETRY 3.1 Describing a Tree 3.2 Fractals 3.3 Dimension 5 The Box-Counting Dimension, 3.4 Statistical Self-Similarity 6 Self-Similarity in Time, 3.5 Fractals and Dynamics 7 Random Walks and Levy Walks 8 Fractal Growth, Sources and Notes Exercises Computer Projects 4 ONE-DIMENSIONAL DIFFERENTIAL EQUATIONS 4.1 Basic Definitions 4.2 Growth and Decay 9 Traffic on the Internet 10 Open Time Histograms in Patch Clamp Experiments" 11 Gompertz Growth of Tumors, 4.3 Multiple Fixed Points 4.4 Geometrical Analysis of One-Dimensional Nonlinear Ordinary Differential Equations 4.5 Algebraic Analysis of Fixed Points 4.6 Differential Equations versus Finite-Difference Equations 4.7 Differential Equations with Inputs 12 Heart Rate Response to Sinusoid Inputs 4.8 Advanced Topic: Time Delays and Chaos 13 Nicholson's Blowflies, Sources and Notes Exercises Computer Projects 5 TWO-DIMENSIONAL DIFFERENTIAL EQUATIONS 5.1 The Harmonic Oscillator 5.2 Solutions, Trajectories, and Flows 5.3 The Two-Dimensional Linear Ordinary Differential Equation 5.4 Coupled First-Order Linear Equations 14 Metastasis of Malignant Tumors, 5.5 The Phase Plane 5.6 Local Stability Analysis of Two-Dimensional, Nonlinear Differential Equations 5.7 Limit Cycles and the van der Pol Oscillator 5.8 Finding Solutions to Nonlinear Differential Equations 15 Action Potentials in Nerve Celis 5.9 Advanced Topic: Dynamics in Three or More Dimensions 5.10 Advanced Topic: Poincar6 Index Theorem Sources and Notes Exercises Computer Projects 6 TIME-SERIES ANALYSIS 6.1 Starting with Data 6.2 Dynamics, Measurements, and Noise 16 Fluctuations in Marine Populations, 6.3 The Mean and Standard Deviation 6.4 Linear Correlations 6.5 Power Spectrum Analysis 17 Daily Oscillations in Zooplankton 6.6 Nonlinear Dynamics and Data Analysis 18 Reconstructing Nerve Cell Dynamics, 6.7 Characterizing Chaos 19 Predicting the Next Ice Age, 6.8 Detecting Chaos and Nonlinearity 6.9 Algorithms and Answers Sources and Notes Exercises Computer Projects APPENDIX A A MULTI-FUNCTIONAL APPENDIX A.1 The Straight Line A.2 The Quadratic Function A.3 The Cubic and Higher-Order Polynomials A.4 The Exponential Function A.5 Sigmoidal Functions A.6 The Sine and Cosine Functions A.7 The Gaussian (or "Normal") Distribution A.8 The Ellipse A.9 The Hyperbola Exercises APPENDIX B A NOTE ON COMPUTER NOTATION SOLUTIONS TO SELECTED EXERCISES BIBLIOGRAPHY INDEX