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2nd Edition

Encounters with Chaos and Fractals




ISBN 9781584885177
Published April 26, 2012 by Chapman and Hall/CRC
387 Pages 159 B/W Illustrations

 
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Book Description

Now with an extensive introduction to fractal geometry

Revised and updated, Encounters with Chaos and Fractals, Second Edition provides an accessible introduction to chaotic dynamics and fractal geometry for readers with a calculus background. It incorporates important mathematical concepts associated with these areas and backs up the definitions and results with motivation, examples, and applications.

Laying the groundwork for later chapters, the text begins with examples of mathematical behavior exhibited by chaotic systems, first in one dimension and then in two and three dimensions. Focusing on fractal geometry, the author goes on to introduce famous infinitely complicated fractals. He analyzes them and explains how to obtain computer renditions of them. The book concludes with the famous Julia sets and the Mandelbrot set.

With more than enough material for a one-semester course, this book gives readers an appreciation of the beauty and diversity of applications of chaotic dynamics and fractal geometry. It shows how these subjects continue to grow within mathematics and in many other disciplines.

Table of Contents

Periodic Points
Iterates of Functions
Fixed Points
Periodic Points
Families of Functions
The Quadratic Family
Bifurcations
Period-3 Points
The Schwarzian Derivative

One-Dimensional Chaos
Chaos
Transitivity and Strong Chaos
Conjugacy
Cantor Sets

Two-Dimensional Chaos
Review of Matrices
Dynamics of Linear Functions
Nonlinear Maps
The Hénon Map
The Horseshoe Map

Systems of Differential Equations
Review of Systems of Differential Equations
Almost Linearity
The Pendulum
The Lorenz System

Introduction to Fractals
Self-Similarity
The Sierpiński Gasket and Other "Monsters"
Space-Filling Curves
Similarity and Capacity Dimensions
Lyapunov Dimension
Calculating Fractal Dimensions of Objects

Creating Fractals Sets
Metric Spaces
The Hausdorff Metric
Contractions and Affine Functions
Iterated Function Systems
Algorithms for Drawing Fractals

Complex Fractals: Julia Sets and the Mandelbrot Set
Complex Numbers and Functions
Julia Sets
The Mandelbrot Set

Computer Programs

Answers to Selected Exercises

References

Index

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Author(s)

Biography

Denny Gulick is a professor in the Department of Mathematics at the University of Maryland. His research interests include operator theory and fractal geometry. He earned a PhD from Yale University.

Reviews

"This text aims to introduce ‘anyone who has a knowledge of calculus’ to ‘chaotic dynamics and fractal geometry at a modest level of sophistication.’ Indeed, the author makes this possible through careful exposition, examples, and exercises …"
—Steve Pederson, Zentralblatt MATH 1253