Discrete chaos: with applications in science and engineering

PREFACE FOREWORD The Stability of One-Dimensional Maps Introduction Maps vs. Difference Equations Maps vs. Differential Equations Linear Maps/Difference Equations Fixed (Equilibrium) Points Graphical Iteration and Stability Criteria for Stability Periodic Points and Their Stability The Period-Doubling Route to Chaos Applications Attraction and Bifurcation Introduction Basin of Attraction of Fixed Points Basin of Attraction of Periodic Orbits Singer's Theorem Bifurcation Sharkovsky's Theorem The Lorenz Map Period-Doubling in the Real World Poincaré Section/Map Appendix Chaos in One Dimension Introduction Density of the Set of Periodic Points Transitivity Sensitive Dependence Definition of Chaos Cantor Sets Symbolic Dynamics Conjugacy Other Notions of Chaos Rössler's Attractor Saturn's Rings Stability of Two-Dimensional Maps Linear Maps vs. Linear Systems Computing An Fundamental Set of Solutions Second-Order Difference Equations Phase Space Diagrams Stability Notions Stability of Linear Systems The Trace-Determinant Plane Liapunov Functions for Nonlinear Maps Linear Systems Revisited Stability via Linearization Applications Appendix Bifurcation and Chaos in Two Dimensions Center Manifolds Bifurcation Hyperbolic Anosov Toral Automorphism Symbolic Dynamics The Horseshoe and Hénon Maps A Case Study: Extinction and Sustainability in Ancient Civilizations Appendix Fractals Examples of Fractals L-System The Dimension of a Fractal Iterated Function System Mathematical Foundation of Fractals The Collage Theorem and Image Compression The Julia and Mandelbrot Sets Introduction 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 Answers to Selected Problems Index.