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