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| Preface | |
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| Author Biography | |
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| List of Symbols | |
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| Foreword | |
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| The Stability of One-Dimensional Maps | |
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| Introduction | |
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| Maps vs. Difference Equations | |
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| Maps vs. Differential Equations | |
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| Euler's Method | |
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| Poincare Map | |
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| Linear Maps/Difference Equations | |
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| Fixed (Equilibrium) Points | |
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| Graphical Iteration and Stability | |
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| Criteria for Stability | |
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| Hyperbolic Fixed Points | |
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| Nonhyperbolic Fixed Points | |
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| Periodic Points and Their Stability | |
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| The Period-Doubling Route to Chaos | |
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| Fixed Points | |
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| 2-Periodic Cycles | |
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| 2[superscript 2]-Periodic Cycles | |
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| Beyond [mu subscript infinity] | |
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| Applications | |
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| Fish Population Modeling | |
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| Attraction and Bifurcation | |
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| Introduction | |
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| Basin of Attraction of Fixed Points | |
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| Basin of Attraction of Periodic Orbits | |
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| Singer's Theorem | |
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| Bifurcation | |
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| Sharkovsky's Theorem | |
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| Li-Yorke Theorem | |
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| A Converse of Sharkovsky's Theorem | |
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| The Lorenz Map | |
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| Period-Doubling in the Real World | |
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| Poincare Section/Map | |
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| Belousov-Zhabotinskii Chemical Reaction | |
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| Appendix | |
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| Chaos in One Dimension | |
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| Introduction | |
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| Density of the Set of Periodic Points | |
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| Transitivity | |
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| Sensitive Dependence | |
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| Definition of Chaos | |
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| Cantor Sets | |
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| Symbolic Dynamics | |
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| Conjugacy | |
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| Other Notions of Chaos | |
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| Rossler's Attractor | |
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| Saturn's Rings | |
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| Stability of Two-Dimensional Maps | |
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| Linear Maps vs. Linear Systems | |
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| Computing A[superscript n] | |
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| Fundamental Set of Solutions | |
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| Second-Order Difference Equations | |
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| Phase Space Diagrams | |
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| Stability Notions | |
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| Stability of Linear Systems | |
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| The Trace-Determinant Plane | |
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| Stability Analysis | |
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| Navigating the Trace-Determinant Plane | |
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| Liapunov Functions for Nonlinear Maps | |
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| Linear Systems Revisited | |
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| Stability via Linearization | |
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| The Hartman-Grobman Theorem | |
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| The Stable Manifold Theorem | |
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| Applications | |
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| The Kicked Rotator and the Henon Map | |
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| The Henon Map | |
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| Discrete Epidemic Model for Gonorrhea | |
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| Perennial Grass | |
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| Appendix | |
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| Bifurcation and Chaos in Two Dimensions | |
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| Center Manifolds | |
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| Bifurcation | |
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| Eigenvalues of 1 or -1 | |
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| A Pair of Eigenvalues of Modulus 1 - The Neimark-Sacker Bifurcation | |
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| Hyperbolic Anosov Toral Automorphism | |
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| Symbolic Dynamics | |
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| Subshifts of Finite Type | |
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| The Horseshoe and Henon Maps | |
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| The Henon Map | |
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| A Case Study: The Extinction and Sustainability in Ancient Civilizations | |
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| Appendix | |
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| Fractals | |
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| Examples of Fractals | |
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| L-system | |
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| The Dimension of a Fractal | |
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| Iterated Function System | |
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| Deterministic IFS | |
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| The Random Iterated Function System and the Chaos Game | |
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| Mathematical Foundation of Fractals | |
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| The Collage Theorem and Image Compression | |
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| The Julia and Mandelbrot Sets | |
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| Introduction | |
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| Mapping by Functions on the Complex Domain | |
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| The Riemann Sphere | |
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| The Julia Set | |
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| Topological Properties of the Julia Set | |
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| Newton's Method in the Complex Plane | |
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| The Mandelbrot Set | |
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| Topological Properties | |
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| Rays and Bulbs | |
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| Rotation Numbers and Farey Addition | |
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| Accuracy of Pictures | |
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| Bibliography | |
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| Answers to Selected Problems | |
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| Index | |