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.

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 Sierpiski Gasket and Other "Monsters"

Space-Filling Curves

Similarity and Capacity Dimensions

Lyapunov Dimensip>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

Author Bio

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.