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 resea...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
New York, NY
Springer New York
1995, 1995
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| Edition: | 1st ed. 1995 |
| Series: | Textbooks in Mathematical Sciences
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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
- 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