Book review

The book provides a complete introduction to the phenomena of dynamical systems for students of mathematics and related sciences. The mathematical tools are restricted to calculus, functions and sets, and the understanding is supported by a large number of experiments and problems. The book covers most of the contemporary results in dynamical systems. First of all images from dynamical systems are shown on 16 pages, beautifully printed in colour. Then, the history of dynamics beginning with Isaac Newton and brought forward by Henry Poincaré is reported. Examples of dynamical systems from finance and ecology are given and modeled by differential equations. Orbits are classified and discussed and it is shown that the computer may lie. The graphical analysis leads to the phase portrait. Fixed and periodic points are investigated. The saddle-node bifurcation and the period-doubling bifurcation prepare the transition to chaos. Symbolic dynamics are presented. The properties of chaos are discussed resulting in Feigenbaum's constant. Then, Sarkovskii's theorem is discussed, critical points and basins of attraction are introduced. The nonconvergence of Newton's method is interpreted and related to chaos. Fractals as geometric objects are used for better understanding of chaos in dynamical systems with many beautiful experiments. Further, dynamical systems in the plane are studied by complex functions. Then, Julia sets and Mandelbrot sets are described in detail. In conclusion further projects and experiments are reported.

The book is addressed to undergraduate students of mathematics but it is also recommended to scientists who are interested in recent contributions of mathematicians to the development of chaotic dynamics without being experienced in mathematical methods. Therefore, it is a worthwhile addition to the general literature on chaos and fractals. The book is very well written including definition, theorems, some proofs and many examples. There is a full index and a good list of references. Moreover, for the experiments and problems of the book software running on Macintosh computers is also available from the publisher. The book should be available in libraries of universities and other institutions of higher learning.