Chaos and Fractals

Zusatztext [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 ... Informationen zum Autor David Feldman joined the faculty at College of the Atlantic in 1998, having completed a PhD in Physics at the University of California. He served as Associate Dean for Academic Affairs from 2003 - 2007. At COA Feldman has taught over twenty different courses in physics, mathematics, and computer science. Feldman's research interests lie in the fields of statistical mechanics and nonlinear dynamics. In his research, he uses both analytic and computational techniques. Feldman has authored research papers in journals including Physical Review E, Chaos, and Advances in Complex Systems. He has recently begun a research project looking at trends in extreme precipitation events in Maine. Klappentext For students with a background in elementary algebra, this book provides a vivid introduction to the key phenomena and ideas of chaos and fractals, including the butterfly effect, strange attractors, fractal dimensions, Julia Sets and the Mandelbrot Set, power laws, and cellular automata. The book includes over 200 end-of-chapter exercises. Zusammenfassung For students with a background in elementary algebra, this book provides a vivid introduction to the key phenomena and ideas of chaos and fractals, including the butterfly effect, strange attractors, fractal dimensions, Julia Sets and the Mandelbrot Set, power laws, and cellular automata. The book includes over 200 end-of-chapter exercises.

Autorentext

David Feldman joined the faculty at College of the Atlantic in 1998, having completed a PhD in Physics at the University of California. He served as Associate Dean for Academic Affairs from 2003 - 2007. At COA Feldman has taught over twenty different courses in physics, mathematics, and computer science.

Feldman's research interests lie in the fields of statistical mechanics and nonlinear dynamics. In his research, he uses both analytic and computational techniques. Feldman has authored research papers in journals including Physical Review E, Chaos, and Advances in Complex Systems. In 2011-12 he was a U.S. Fulbright Lecturer in Kigali, Rwanda.

Klappentext

For students with a background in elementary algebra, this book provides a vivid introduction to the key phenomena and ideas of chaos and fractals, including the butterfly effect, strange attractors, fractal dimensions, Julia Sets and the Mandelbrot Set, power laws, and cellular automata. The book includes over 200 end-of-chapter exercises.

Zusammenfassung
This book provides the reader with an elementary introduction to chaos and fractals, suitable for students with a background in elementary algebra, without assuming prior coursework in calculus or physics. It introduces the key phenomena of chaos - aperiodicity, sensitive dependence on initial conditions, bifurcations - via simple iterated functions. Fractals are introduced as self-similar geometric objects and analyzed with the self-similarity and box-counting dimensions. After a brief discussion of power laws, subsequent chapters explore Julia Sets and the Mandelbrot Set. The last part of the book examines two-dimensional dynamical systems, strange attractors, cellular automata, and chaotic differential equations. The book is richly illustrated and includes over 200 end-of-chapter exercises. A flexible format and a clear and succinct writing style make it a good choice for introductory courses in chaos and fractals.

Inhalt

• I. Introducing Discrete Dynamical Systems

• 0: Opening Remarks

• 1: Functions

• 2: Iterating Functions

• 3: Qualitative Dynamics

• 4: Time Series Plots

• 5: Graphical Iteration

• 6: Iterating Linear Functions

• 7: Population Models

• 8: Newton, Laplace, and Determinism

• II. Chaos

• 9: Chaos and the Logistic Equation

• 10: The Buttery Effect

• 11: The Bifurcation Diagram

• 12: Universality

• 13: Statistical Stability of Chaos

• 14: Determinism, Randomness, and Nonlinearity

• III. Fractals

• 15: Introducing Fractals

• 16: Dimensions

• 17: Random Fractals

• 18: The Box-Counting Dimension

• 19: When do Averages exist?

• 20: Power Laws and Long Tails

• 20: Introducing Julia Sets

• 21: Infinities, Big and Small

• IV. Julia Sets and The Mandelbrot Set

• 22: Introducing Julia Sets

• 23: Complex Numbers

• 24: Julia Sets for f(z) = z2 + c

• 25: The Mandelbrot Set

• V. Higher-Dimensional Systems

• 26: Two-Dimensional Discrete Dynamical Systems

• 27: Cellular Automata

• 28: Introduction to Differential Equations

• 29: One-Dimensional Differential Equations

• 30: Two-Dimensional Differential Equations

• 31: Chaotic Differential Equations and Strange Attractors

• VI. Conclusion

• 32: Conclusion

• VII. Appendices

• A: Review of Selected Topics from Algebra

• B: Histograms and Distributions

• C: Suggestions for Further Reading