Chaos and Fractals

An Elementary Introduction
Erscheinungsjahr 2012
ISBN: 978-0-19-956643-3
Verlag: Oxford University Press

Buch, Englisch, 432 Seiten, Format (B × H): 195 mm x 248 mm, Gewicht: 1098 g

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.

Zielgruppe

Undergraduate students and lecturers on specialist and non-specialist courses in physics and mathematics.

Autoren/Hrsg.

Feldman, David P.

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Dynamische Systeme
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Numerische Mathematik
• Naturwissenschaften Physik Angewandte Physik Statistische Physik, Dynamische Systeme

Inhaltsverzeichnis

- 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

Über Autor(innen)

Feldman, David P.
David P. Feldman, Department od Physics and Mathematics, College of the Atlantic, Bar Harbor, Maine, USA

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.

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.