Understanding Nonlinear Dynamics
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
1 online resource (xx, 420 pages)
9781461208235, 1461208238
853263677
Print version:
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
6.5 Power 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
English