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## Details

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. |

### Abstract:

## Reviews

*Editorial reviews*

Publisher Synopsis

[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 accessible ... * Danny Yee, Danny Reviews * The book is very well produced, with excellent diagrams and very informative notes provided beside the main text. It also provides an extensive list of references for further reading. * Scottish Mathematical Council Journal * For the right audience and instructor, this is a wonderful book. With considerable effort on both sides it can take a wide audience with modest mathematics to a reasonable understanding of what is behind much of the complex phenomena seen in modern mathematical models of the physical universe. * Thomas B. Ward, Durham University * David P. Feldman provides a delightful and thoughtful introduction to chaos and fractals requiring only a good background in algebra. The formal treatment of nonlinear dynamics, chaotic behavior, Lyapunov exponents, and fractal dimensions is leavened with creative analogies and many helpful and visually attractive figures and diagrams. Even more mathematically sophisticated readers will find this book a good starting point in exploring the complex and beguilingrealms of chaos and fractals. * Robert C. Hilborn, Associate Executive Officer, American Association of Physics Teachers * The style of writing is easy on the reader. The explanations are clear and illustrated with many diagrams and side notes....[Feldman] has produced an excellent book. * John Sykes, Mathematics in School * Chaos and fractals are two intertwined concepts that have revolutionized many areas of science and renewed popular interest in mathematics over the past few decades. Feldman's book is a rich resource for anyone who wants a deeper understanding of these subjects without the need for advanced mathematics. * Julien Clinton Sprott, University of Wisconsin-Madison * The only textbook on chaos and fractals for non-science and mathematics majors. Covers central phenomena and ideas of chaos and fractals in a careful, intellectually honest, but accessible way. * L'Enseignement Mathematique (2) 59 * This is an excellent book, and is highly recommended. * Mark Hunacek, Mathematical Association of America * For the right audience and instructor, this is a wonderful book. With considerable effort on both sides it can take a wide audience with modest mathematics to a reasonable understanding of what is behind much of the complex phenomena seen in modern mathematical models of the physical universe. * Thomas B. Ward, Zentralblatt MATH * ... obtains top marks ... For any lecturer or teacher looking for a text on these subjects, this book is worthy of your consideration. * Gazette of the Australian Mathematical Society * Read more...

*User-contributed reviews*