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An Introduction to Difference Equations
Contents
- The Stability of One-Dimensional Maps
- Maps Versus Difference Equations
- Maps Versus Differential Equations
- Linear Maps/Difference Equations
- Fixed Points
- Graphical Iteration and Stability
- Criteria for Stability
- Periodic Points and Their Stability
- The Period-Doubling Route to Chaos
- Applications
- Sharkovsky's Theorem and Bifurcation
- The Mystery of Period 3
- Converse of Sharkovsky's Theorem
- Basin of Attraction
- The Schwarzian Derivative
- Bifurcation
- The Lorenz Map
- Chaos in One Dimension
- Introduction
- Metric Spaces
- Transitivity
- Sensitive Dependence and Liapunov Exponents
- Definition and Chaos
- Symbolic Dynamics
- Conjugacy
- Stability of Two-Dimensional Maps
- Linear Maps Versus Linear Systems
- Computing An
- Phase Space
- Liapunov Functions for Nonlinear Maps
- Linear Systems Revisited
- Stability via Linearization
- Applications
- Chaos in Two Dimensions
- Hyperbolic Anosov Toral Automorphism
- Symbolic Dynamics
- The Horseshoe and HTnon Maps
- Center Manifolds
- Bifurcation
- Fractals
- Examples of Fractals
- The Dimension of Fractal
- Iterated Function System
- Mathematical Foundation of Fractals
- The Collage Theorem and Image Compression
- The Julia and Mandelbrot Sets
- Mapping by Functions on the Complex Domain
- The Riemann Sphere
- The Julia Set
- Topological Properties of the Julia Set
- Newton's Method in the Complex Plane
- The Mandelbrot Set
- Bibliography
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