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Chaos and fractals : an elementary introduction

Author: David P Feldman
Publisher: Oxford : Oxford University Press, 2012.
Edition/Format:   eBook : Document : English : First editionView all editions and formats
Database:WorldCat
Summary:
For students with a background in elementary algebra, this text provides a vivid introduction to the key phenomena and ideas of chaos and fractals, including the butterfly effect, strange attractors, fractal dimensions, Julia sets and the Mandelbrot set, power laws, and cellular automata.
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Additional Physical Format: Version imprimée:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: David P Feldman
ISBN: 9781283643887 128364388X 9780191637520 0191637521 6613956384 9786613956385
OCLC Number: 870092468
Description: 1 online resource (1 ressource en ligne (xxi, 408 pages)) : illustrations
Contents: Cover; Contents; I: Introducing Discrete Dynamical Systems; 0 Opening Remarks; 0.1 Chaos; 0.2 Fractals; 0.3 The Character of Chaos and Fractals; 1 Functions; 1.1 Functions as Actions; 1.2 Functions as a Formula; 1.3 Functions are Deterministic; 1.4 Functions as Graphs; 1.5 Functions as Maps; Exercises; 2 Iterating Functions; 2.1 The Idea of Iteration; 2.2 Some Vocabulary and Notation; 2.3 Iterated Function Notation; 2.4 Algebraic Expressions for Iterated Functions; 2.5 Why Iteration?; Exercises; 3 Qualitative Dynamics: The Fate of the Orbit; 3.1 Dynamical Systems. 3.2 Dynamics of the Squaring Function3.3 The Phase Line; 3.4 Fixed Points via Algebra; 3.5 Fixed Points Graphically; 3.6 Types of Fixed Points; Exercises; 4 Time Series Plots; 4.1 Examples of Time Series Plots; Exercises; 5 Graphical Iteration; 5.1 An Initial Example; 5.2 The Method of Graphical Iteration; 5.3 Further Examples; Exercises; 6 Iterating Linear Functions; 6.1 A Series of Examples; 6.2 Slopes of +1 or -1; Exercises; 7 Population Models; 7.1 Exponential Growth; 7.2 Modifying the Exponential Growth Model; 7.3 The Logistic Equation; 7.4 A Note on the Importance of Stability. 7.5 Other r ValuesExercises; 8 Newton, Laplace, and Determinism; 8.1 Newton and Universal Mechanics; 8.2 The Enlightenment and Optimism; 8.3 Causality and Laplace's Demon; 8.4 Science Today; 8.5 A Look Ahead; II: Chaos; 9 Chaos and the Logistic Equation; 9.1 Periodic Behavior; 9.2 Aperiodic Behavior; 9.3 Chaos Defined; 9.4 Implications of Aperiodic Behavior; Exercises; 10 The Butterfly Effect; 10.1 Stable Periodic Behavior; 10.2 Sensitive Dependence on Initial Conditions; 10.3 SDIC Defined; 10.4 Lyapunov Exponents; 10.5 Stretching and Folding: Ingredients for Chaos. 10.6 Chaotic Numerics: The Shadowing LemmaExercises; 11 The Bifurcation Diagram; 11.1 A Collection of Final-State Diagrams; 11.2 Periodic Windows; 11.3 Bifurcation Diagram Summary; Exercises; 12 Universality; 12.1 Bifurcation Diagrams for Other Functions; 12.2 Universality of Period Doubling; 12.3 Physical Consequences of Universality; 12.4 Renormalization and Universality; 12.5 How are Higher-Dimensional Phenomena Universal?; Exercises; 13 Statistical Stability of Chaos; 13.1 Histograms of Periodic Orbits; 13.2 Histograms of Chaotic Orbits; 13.3 Ergodicity; 13.4 Predictable Unpredictability. Exercises14 Determinism, Randomness, and Nonlinearity; 14.1 Symbolic Dynamics; 14.2 Chaotic Systems as Sources of Randomness; 14.3 Randomness?; 14.4 Linearity, Nonlinearity, and Reductionism; 14.5 Summary and a Look Ahead; Exercises; III: Fractals; 15 Introducing Fractals; 15.1 Shapes; 15.2 Self-Similarity; 15.3 Typical Size?; 15.4 Mathematical vs. Real Fractals; Exercises; 16 Dimensions; 16.1 How Many Little Things Fit inside a Big Thing?; 16.2 The Dimension of the Snowflake; 16.3 What does D {u2248} 1.46497 Mean?; 16.4 The Dimension of the Cantor Set; 16.5 The Dimension of the Sierpiński Triangle.
Responsibility: David P. Feldman, College of the Atlantic, Bar Harbor, Maine, USA.

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For students with a background in elementary algebra, this book provides a vivid introduction to the key phenomena and ideas of chaos and fractals, including the butterfly effect, strange attractors,  Read more...

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[haos and Fractals] offers at least the possibility of a radically different trajectory for school teaching, providing a motivated pathway to a lot of fascinating mathematics not normally considered Read more...

 
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