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Understanding Nonlinear Dynamics

by Leon Glass

  • ISBN: 9780387944401
  • ISBN10: 0387944400

Understanding Nonlinear Dynamics

by Leon Glass

  • List Price: $84.95
  • Binding: Paperback
  • Publisher: Springer Verlag
  • Publish date: 04/01/1995
  • ISBN: 9780387944401
  • ISBN10: 0387944400
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Description: 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 Lvy 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: Poincar 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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