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
Iterates of Functions
Families of Functions
The Quadratic Family
The Schwarzian Derivative
Transitivity and Strong Chaos
Review of Matrices
Dynamics of Linear Functions
The Hénon Map
The Horseshoe Map
Systems of Differential Equations
Review of Systems of Differential Equations
The Lorenz System
Introduction to Fractals
The Sierpiński Gasket and Other "Monsters"
Similarity and Capacity Dimensions
Calculating Fractal Dimensions of Objects
Creating Fractals Sets
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
The Mandelbrot Set
Answers to Selected Exercises
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.
"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