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Fractals and Chaos
An illustrated course




ISBN 9780750304009
Published January 1, 1997 by CRC Press
256 Pages

 
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Book Description

Fractals and Chaos: An Illustrated Course provides you with a practical, elementary introduction to fractal geometry and chaotic dynamics-subjects that have attracted immense interest throughout the scientific and engineering disciplines. The book may be used in part or as a whole to form an introductory course in either or both subject areas. A prominent feature of the book is the use of many illustrations to convey the concepts required for comprehension of the subject. In addition, plenty of problems are provided to test understanding. Advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas. The book can be used as a text for undergraduate courses or for self-study.

Table of Contents

INTRODUCTION
Introduction
A matter of fractals
Deterministic chaos
Chapter summary and further reading

REGULAR FRACTALS AND SELF-SIMILARITY
Introduction
The Cantor set
Non-fractal dimensions: the Euclidean and topological dimension
The similarity dimension
The Koch curve
The quadratic Koch curve
The Koch island
Curves in the plane with similarity dimension exceeding 2
The Sierpinski gasket and carpet
The Menger Sponge
Chapter summary and further reading
Revision questions and further tasks

RANDOM FRACTALS
Introduction
Randomizing the Cantor set and Koch curve
Fractal boundaries
The box counting dimension and the Hausdorff dimension
The structured walk technique and the divider dimension
The perimeter-area relationship
Chapter summary and further reading
Revision questions and further tasks

REGULAR AND FRACTIONAL BROWNIAN MOTION
Introduction
Regular Brownian motion
Fractional Brownian motion: time traces
Fractional Brownian surfaces
Fractional Brownian motion: spatial trajectories
Diffusion limited aggregation
The color and power and noise
Chapter summary and further reading
Revision questions and further tasks

ITERATIVE FEEDBACK PROCESSES AND CHAOS
Introduction
Population growth and the Verhulst model
The logistic map
The effect of variation in the control parameter
General form of the iterated solutions of the logistic map
Graphical iteration of the logistic map
Bifurcation, stability and the Feigenbaum number
A two dimensional map: the Henon model
Iterations in the complex plane: Julia sets and the Mandelbrot set
Chapter summary and further reading
Revision questions and further tasks

CHAOTIC OSCILLATIONS
Introduction
A simple nonlinear mechanical oscillator: the Duffing oscillator
Chaos in the weather: the Lorenz model
The Rossler systems
Phase space, dimension and attractor form
Spatially extended systems: coupled oscillators
Spatially extended systems: fluids
Mathematical routes to chaos and turbulence
Chapter summary and further reading
Revision questions and further tasks

CHARACTERIZING CHAOS
Introduction
Preliminary characterization: visual inspection
Preliminary characterization: frequency spectra
Characterizing chaos: Lyapunov exponents
Characterizing chaos: dimension estimates
Attractor reconstruction
The embedding dimension for attractor reconstruction
The effect of noise
Regions of behavior on the attractor and characterization limitations
Chapter summary and further reading
Revision questions and further task

APPENDIX 1: Computer Program for Lorenz Equations
APPENDIX 2: Illustrative Papers
APPENDIX 3: Experimental Chaos

SOLUTIONS
REFERENCES

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Reviews

"Fractals and Chaos: An Illustrated Course is well designed for self-study, making it a great practical resource for those working in the physical sciences or engineering as well as for students."
—Danny Yee’s Book Reviews, February 2016