Discrete chaos.

*(English)*Zbl 0945.37010
Boca Raton, FL: Chapman & Hall/CRC. xiii, 355 p. (2000).

The last three decades witnessed a surge of research activities in deterministic chaos theory and fractals. So numerous books on these subjects have appeared, especially in the last decade. The book under review grew out of a course on discrete dynamical systems/difference equations taught by the author at the Trinity University (San Antonio, Texas), since 1992. The book intends “to give a thorough exposition of stability theory in one and two dimensions including the method of Lyapunov”. In other words the subject matter is according to the taste of the author. The same holds for the organization and the structure of the book.

Ch. 1 is about the stability of one dimensional maps. So it is surprising that without introducing the term of chaos the part 1.8. is titled “The period doubling route to chaos”. Interestingly enough the definition of chaos is given only in part 3.5 (p. 117)!

Ch. 2 is named “Sharkovsky’s theorem and bifurcation”. But it does not mean that one speaks about Sharkovsky’s bifurcation as such, but about bifurcations of 1-dim maps. Unfortunately, the author does not present a classification of such possible bifurcations. It is funny that the Schwarzian derivative is defined in 1.6.2 (Def. 1.3, p. 24) but also in 2.4. In this chapter the author cites the famous result of Li and Yorke “Period three implies chaos”, but unfortunately does not mention there the first rigorous definition of chaos. Only in Ch. 3 chaos is defined in the 1-dim case. But there the term chaos is defined according to the one introduced by R. L. Devaney in his book “An introduction to chaotic dynamical systems” (Addison-Wesley) of 1989 (!) (see Zbl 0695.58002). However, this is not a very fortunate way as it is shown later that three conditions for a map \(f\) to be chaotic, namely \(f\) is transitive; the set of periodic points of \(f\) is dense; and the last \(f\) has sensitive dependence on initial conditions, are interdependent.

Ch. 4 is mostly about the stability of 2-dim linear maps. Here also the Lyapunov function method for nonlinear maps is introduced. The next Ch. 5, named “Chaos in two dimensions” brings examples of chaotic dynamics, namely hyperbolic Anosov toral automorphism, Smale’s horseshoe and the Hénon map. Here again, one problem, namely the Hénon map is treated twice. It was already mentioned in Ch. 4 as example of a strange attractor. Again, the term ‘strange attractor’ has not been introduced in the book at all! Besides, in 5.3.1. the author says “computer iteration shows the existence of strange attractors, however, the proof of this statement continues to elude the mathematicians”. Unfortunately the author does not point out what is behind it, namely that the Hénon map is dissipative.

The last two chapters 6 and 7 are devoted to fractals and Julia and Mandelbrot sets. The treatment of fractals at the end of the book is also surprising as chaotic attractors possess in a sense a fractal structure and the fractal dimension is an important characteristic of chaotic attractors.

As mentioned above, the organization of the book is of a very personal taste. Even the diagram showing the interdependence of chapters (pp. xi) belongs to this category. The book is recommended by the author for two semester courses or some selection of chapters for one semester courses. But it is not specified for which courses and for which part (in time) of them.

On the other hand the book is endowed with many practical exercises, including solutions. But some of them are formulated as a part of the main text. So to understand the interpretation one needs to go through the exercises, as well.

The style is almost elementary and the only prerequisites for the course or following the book is, according to the author, calculus and linear algebra.

One must respect the graphics of the book are very good, without any technical errors. The references are unfortunately not complete and not up-to-date.

So one can recommend this book only as complementary for introductory courses on discrete chaos and fractals but not so much for such courses on discrete dynamical systems and difference equations, in general.

Ch. 1 is about the stability of one dimensional maps. So it is surprising that without introducing the term of chaos the part 1.8. is titled “The period doubling route to chaos”. Interestingly enough the definition of chaos is given only in part 3.5 (p. 117)!

Ch. 2 is named “Sharkovsky’s theorem and bifurcation”. But it does not mean that one speaks about Sharkovsky’s bifurcation as such, but about bifurcations of 1-dim maps. Unfortunately, the author does not present a classification of such possible bifurcations. It is funny that the Schwarzian derivative is defined in 1.6.2 (Def. 1.3, p. 24) but also in 2.4. In this chapter the author cites the famous result of Li and Yorke “Period three implies chaos”, but unfortunately does not mention there the first rigorous definition of chaos. Only in Ch. 3 chaos is defined in the 1-dim case. But there the term chaos is defined according to the one introduced by R. L. Devaney in his book “An introduction to chaotic dynamical systems” (Addison-Wesley) of 1989 (!) (see Zbl 0695.58002). However, this is not a very fortunate way as it is shown later that three conditions for a map \(f\) to be chaotic, namely \(f\) is transitive; the set of periodic points of \(f\) is dense; and the last \(f\) has sensitive dependence on initial conditions, are interdependent.

Ch. 4 is mostly about the stability of 2-dim linear maps. Here also the Lyapunov function method for nonlinear maps is introduced. The next Ch. 5, named “Chaos in two dimensions” brings examples of chaotic dynamics, namely hyperbolic Anosov toral automorphism, Smale’s horseshoe and the Hénon map. Here again, one problem, namely the Hénon map is treated twice. It was already mentioned in Ch. 4 as example of a strange attractor. Again, the term ‘strange attractor’ has not been introduced in the book at all! Besides, in 5.3.1. the author says “computer iteration shows the existence of strange attractors, however, the proof of this statement continues to elude the mathematicians”. Unfortunately the author does not point out what is behind it, namely that the Hénon map is dissipative.

The last two chapters 6 and 7 are devoted to fractals and Julia and Mandelbrot sets. The treatment of fractals at the end of the book is also surprising as chaotic attractors possess in a sense a fractal structure and the fractal dimension is an important characteristic of chaotic attractors.

As mentioned above, the organization of the book is of a very personal taste. Even the diagram showing the interdependence of chapters (pp. xi) belongs to this category. The book is recommended by the author for two semester courses or some selection of chapters for one semester courses. But it is not specified for which courses and for which part (in time) of them.

On the other hand the book is endowed with many practical exercises, including solutions. But some of them are formulated as a part of the main text. So to understand the interpretation one needs to go through the exercises, as well.

The style is almost elementary and the only prerequisites for the course or following the book is, according to the author, calculus and linear algebra.

One must respect the graphics of the book are very good, without any technical errors. The references are unfortunately not complete and not up-to-date.

So one can recommend this book only as complementary for introductory courses on discrete chaos and fractals but not so much for such courses on discrete dynamical systems and difference equations, in general.

Reviewer: L.Andrey (Praha)

##### MSC:

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37G10 | Bifurcations of singular points in dynamical systems |

37G05 | Normal forms for dynamical systems |

39A11 | Stability of difference equations (MSC2000) |

37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |

37E05 | Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth) |