Understanding Nonlinear Dynamics

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics ( TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathematical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. About the Authors Daniel Kaplan specializes in the analysis of data using techniques motivated by nonlinear dynamics. His primary interest is in the interpretation of irregular physiological rhythms, but the methods he has developed have been used in geo physics, economics, marine ecology, and other fields. He joined McGill in 1991, after receiving his Ph.D from Harvard University and working at MIT. His un dergraduate studies were completed at Swarthmore College. He has worked with several instrumentation companies to develop novel types of medical monitors.

Klappentext

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. Topics include finite difference equations, the concept of chaos, networks, cellular automata, one- 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 for each chapter, and basic mathematical reviews are included in an appendix. Popular mathematical concepts in dynamical systems are discussed, each in the context of applied examples from biological systems. The book is based on a course taught by the authors to students in biology and pre-med programs. Requires one year of calculus.

Inhalt
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.- 2 Boolean Networks and Cellular Automata.- 2.1 Elements and Networks.- 2.2 Boolean Variables, Functions, and Networks.- 2.3 Boolean Functions and Biochemistry.- 2.4 Random Boolean Networks.- 2.5 Cellular Automata.- 2.6 Advanced Topic: Evolution and Computation.- 3 Self-Similarity and Fractal Geometry.- 3.1 Describing a Tree.- 3.2 Fractals.- 3.3 Dimension.- 3.4 Statistical Self-Similarity.- 3.5 Fractals and Dynamics.- 4 One-Dimensional Differential Equations.- 4.1 Basic Definitions.- 4.2 Growth and Decay.- 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.- 4.8 Advanced Topic: Time Delays and Chaos.- 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.- 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.- 5.9 Advanced Topic: Dynamics in Three or More Dimensions.- 5.10 Advanced Topic: Poincaré Index Theorem.- 6 Time-Series Analysis.- 6.1 Starting with Data.- 6.2 Dynamics, Measurements, and Noise.- 6.3 The Mean and Standard Deviation.- 6.4 Linear Correlations.- 6.5Power Spectrum Analysis.- 6.6 Nonlinear Dynamics and Data Analysis.- 6.7 Characterizing Chaos.- 6.8 Detecting Chaos and Nonlinearity.- 6.9 Algorithms and Answers.- 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.