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Understanding Nonlinear Dynamics / Edition 1

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Overview

This book presents the main concepts and applications of nonlinear dynamics at an elementary level. The book is based on a one-semester undergraduate course that has been given since 1975 at McGill University and has been constantly updated to keep up with current developments. Based on the authors' successful course for undergraduate students in the biological sciences, the primer presents the main concepts of non-linear dynamics at a level requiring only one year of calculus. This text will appeal to courses being offered in both mathematics and biology. Topics include finite difference equations, the concept of chaos, networks, cellular automata, on- and two-dimensional differential equations, the dynamics of non-linear equations, and linear stability analysis. Examples are all from the biological sciences, exercises are included in each chapter, and basic mathematical reviews are included in an appendix.

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Product Details

  • ISBN-13: 9780387944234
  • Publisher: Springer New York
  • Publication date: 4/13/1995
  • Series: Texts in Applied Mathematics Series , #19
  • Edition description: 1995
  • Edition number: 1
  • Pages: 420
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.00 (d)

Table of Contents

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 Salamanders.- 2.3 Boolean Functions and Biochemistry.- 2.4 Random Boolean Networks.- 2.5 Cellular Automata.- 4 Spiral Waves in Chemistry and Biology.- 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 Lévy 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 Cells.- 5.9 Advanced Topic: Dynamics in Three or More Dimensions.- 5.10 Advanced Topic: Poincaré 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.

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