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Encounters with chaos and fractals. 2nd ed. (English) Zbl 1253.37001

Boca Raton, FL: CRC Press (ISBN 978-1-58488-517-7/hbk). xvi, 371 p. (2012).
This text aims to introduce “anyone who has a knowledge of calculus” to “chaotic dynamics and fractal geometry at a modest level of sophistication.” Indeed, the author makes this possible through careful exposition, examples, and exercises presenting topics such as the Smale horseshoe map, the Lorenz system, dimension, and iterated function systems. Along the way the author familiarizes the student reader with more advanced background material such as the Cantor set, space-filling curves, Cauchy sequences, complete metric spaces, the Bolzano-Weierstrass theorem, and more.
The text is divided into the following chapters: 1. Periodic points; 2. One-dimensional chaos; 3. Two-dimensional chaos; 4. Systems of differential equations; 5. Introduction to fractals;
6. Creating fractal sets; 7. Complex fractals: Julia sets and the Mandelbrot set. {}Chapters 1–4 correspond to Chapters 1–3 and 5 of the original edition from 1992, while Chapters 5, 6, and 7 expand on Chapter 4 (“Fractals”) of the original edition. The Appendix contains ten computer programs for MATLAB (as opposed to TRUE BASIC in the original edition) including programs to produce the logistic family bifurcation diagram, the Hénon attractor, Julia sets, the Mandelbrot set, and the Barnsley fern leaf.

MSC:

37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
37-04 Software, source code, etc. for problems pertaining to dynamical systems and ergodic theory
28-04 Software, source code, etc. for problems pertaining to measure and integration
37E05 Dynamical systems involving maps of the interval
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C10 Dynamics induced by flows and semiflows
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
28A80 Fractals

Software:

Matlab
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