Encounters with Chaos and Fractals, Third Edition provides an accessible introduction to chaotic dynamics and fractal geometry. It incorporates important mathematical concepts and backs up the definitions and results with motivation, examples, and applications.
The third edition updates this classic book for a modern audience. New applications on contemporary topics, like data science and mathematical modeling, appear throughout. Coding activities are transitioned to open-source programming languages, including Python.
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 authors introduce famous, infinitely complicated fractals. How to obtain computer renditions of them is explained. The book concludes with Julia sets and the Mandelbrot set.
The Third Edition includes:
- More coding activities incorporated in each section with expanded code to include pseudo-code, with specific examples in MATLAB® (or its open-source cousin Octave) and Python
- Additional exercises—many updated—from previous editions
- Proof-writing exercises for a more theoretical course
- Revised sections to include historical context
- Short sections added to explain applied problems in developing mathematics
This edition reveals how these ideas are continuing to be applied in the 21st century, while connecting to the long and winding history of dynamical systems. The primary focus is the beauty and diversity of these ideas. Offering more than enough material for a one-semester course, the authors show how these subjects continue to grow within mathematics and in many other disciplines.
Chapter 1 Periodic Points
Iterates of Functions
Graphical Analysis of Iterates
Section 1.1 Exercises
Fixed Points
Attracting and Repelling Fixed Points
Basins of Attraction
Eventually Fixed Points
Section 1.2 Exercises
Periodic Points
Attracting Periodic Points
Time Series and Periodic Points
Section 1.3 exercises
Families of Functions
The Family {gμ}
The Tent Family {Tμ}
Eventually Periodic and Periodic Points of T
Section 1.4 Exercises
The Quadratic Family
Section 1.5 Exercises
Bifurcation Diagrams
Period-Doubling Bifurcations
Tangent Bifurcations
Section 1.6 Exercises
Period-3 Points
Section 1.7 Exercises
The Schwarzian Derivative
Section 1.8 Exercises
Chapter 2 One-Dimensional Chaos
Chaos
Sensitive Dependence on Initial Conditions
Lyapunov Exponents
Chaos
The Butterfly Effect
The Asteroid Belt
Conclusion
Section 2.1 Exercises
Transitivity and Strong Chaos
Strong Chaos
Section 2.2 Exercises
Conjugacy
Section 2.3 Exercises
Cantor Sets
The Cantor Ternary Set
Strong Chaos of Functions in {Qμ}
Section 2.4 Exercises
Chapter 3 Two-Dimensional Chaos
Review of Matrices
Brief Review of 2 × 2 Matrices
Similar Matrices
Section 3.1 Exercises
Dynamics of Linear Functions
Linear Functions
Dynamics of Linear Functions
Section 3.2 Exercises
Nonlinear Maps
Baker’s Functions
Section 3.3 Exercises
The H ́enon Map
Section 3.4 Exercises
The Horseshoe Map
Homoclinic Points
The Williams Solenoid
Chapter 4 Systems of Differential Equations
Review of Systems of Differential Equations
Linear Differential Equations
Systems of Two Linear Differential Equations
Table of results illustrating the relationship between eigenvalues and critical points
Exercises 4.1
Almost Linearity
Limit Cycles
Exercises 4.2
The Pendulum
Exercises 4.3
The Lorenz System
Conclusion
Exercises 4.4
Chapter 5 Introduction to Fractals
Self-Similarity
The Cantor Set Revisited
The Length of the Cantor Set
The Devil’s Staircase
Exercies 5.1
The Sierpiński Gasket and Other “Monsters”
The Chaos Game
The Sierpiński Carpet
The Menger Sponge
The Koch Curve
The Koch Snowflake
Exercises 5.2
Space-Filling Curves
Hilbert’s Space-Filling Curve
Cauchy Sequences
Exercise 5.3
Similarity and Capacity Dimensions
Similarity Dimension
Capacity Dimension
Exercises 5.4
Lyapunov Dimension
Exercises 5.5
Calculating Fractal Dimensions of Objects
The Compass Dimension
Exercises 5.6
Chapter 6 Creating Fractal Sets
Metric Spaces
Complete Metric Spaces
Exercises 6.1
The Hausdorff Metric
Exercises 6.2
Contractions and Affine Functions
Contractions .
Affine Functions
Isometries
Exercises 6.3
Iterated Function Systems
Exercises 6.4
Algorithms for Drawing Fractals
The Complete Iteration Algorithm
The Random Iteration Algorithm
The Chaos Game Revisited
Exercises 6.5
Chapter 7 Complex Fractals: Julia Sets and The Mandelbrot
Complex Numbers and Functions
Complex Functions
Zeros and Fixed Points of Complex Functions
Periodic Points
Attracting and Repelling Periodic Points
Exercises 7.1
Julia Sets
Exercises 7.2
The Mandelbrot Set
Exercises 7.3
APPENDIX: COMPUTER PROGRAMS
ANSWERS TO SELECTED EXERCISES
Biography
Denny Gulick is Professor Emeritus 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.
Jeff Ford is a Visiting Assistant Professor of Mathematics at Gustavus Adolphus College. He earned his Bachelor’s degree from Gustavus Adolphus College, his Master’s degree in mathematics from Minnesota State University-Mankato, and his Ph.D. in mathematics from Auburn University, studying under Dr. Krystyna Kuperberg. Jeff is interested in the existence of volume-preserving dynamical systems with unique properties. Jeff uses and assesses a variety of active learning techniques in his class including inquiry-based learning and team-based learning. His scholarship in this area centers on understanding how active learning techniques improve confidence and reduce anxiety in undergraduate students.
