UK Nonlinear News, May 1998

Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics

By A. Lasota and M.C. Mackey

Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension

By A. Boyarsky and P. Góra

Reviewed by Dr. Jaroslav Stark

The idea that simple deterministic systems can generate complex, random looking behaviour is now well established in many scientific fields. By far the most familiar way of studying such systems is using topological and geometric techniques which were pioneered by Poincaré at the end of the last century. These typically focus on objects in the state space such as periodic orbits, stable and unstable manifolds and invariant fractal sets such as horseshoes or strange attractors. Such methods have generated a profusion of important results and provided substantial insights into the behaviour of many classes of nonlinear dynamical systems. However, they perhaps suffer from one significant weakness, namely that they concentrate too much on detailed properties which are often of little relevance to practical applications, and in any case are destroyed by the addition of even the slightest amount of noise.

Such considerations prompt a different approach to nonlinear systems, making use of probabilistic techniques and seeking to study whole ensembles of trajectories, rather than just single orbits. This approach dates back nearly as far as the geometric point of view, namely at least to the work of G.D. Birkhoff in the 1930's. Furthermore, under the name of ergodic theory, there is a large body of beautiful and important results in this area. On the whole, however, this has received attention more from pure mathematicians than from applied scientists and engineers (one exception is Lyapunov exponents, which are widely used in many applications, but whose definition relies on a deep theorem from ergodic theory).

The two books under review focus on a particular subset of probabilistic techniques, namely those using Frobenius-Perron operators to study invariant densities. Informally speaking, a density gives the probability of finding an orbit in different parts of the state space at a given time. The time evolution of densities under the action of a given dynamical system is given by the Frobenius-Perron operator for that system. Invariant densities are fixed points of this operator and thus describe the asymptotic statistical behaviour of the system; they are analogous invariant sets in the geometric approach.

The first book, Chaos, Fractals, and Noise , published by Springer-Verlag in 1994, is a revised edition of the classic Probabilistic Properties of Deterministic Systems , which first appeared in 1985, published by CUP. This has been the standard reference work in this area during the intervening years, and the revision only serves to strengthens this position. Laws of Chaos on the other hand is a much more specialised volume, published by Birkhäuser last year.

Both books are organised in a similar fashion. Following an introduction, they give a fairly comprehensive account of necessary prerequisites in measure theory, ergodic theory and functional analysis. This makes both books reasonably self contained, and additional definitions and explanations of mathematical concepts are given as and when required. However, despite this, I suspect than non-mathematicians will find both books heavy going in places (particularly so in the case of Laws of Chaos ).

Both books then present the core material on densities and Frobenius-Perron operators. Here Laws of Chaos takes a much more restricted approach, concentrating on a class of transformations of the interval, whilst Chaos, Fractals, and Noise develops these ideas in the context of general deterministic mappings. On the other hand, Laws of Chaos develops the material further and hence provides a useful review of the current state of this field for the specialist reader. It then continues with a single chapter on perturbations, both deterministic and stochastic, a brief chapter discussing the inverse problem of determining a dynamical system if one knows its invariant density and finally concludes with a detailed description of six applications. The last part of the book consists of some 70 pages of solutions and hints to exercises. This will be a great bonus for anyone contemplating teaching a course on this material.

Chaos, Fractals, and Noise on the other hand branches out in several different directions once it has dealt with the basic material and hence gives an account of a much wider range of topics. First it extends the basic concepts to continuous time systems, and then discusses the relation between discrete and continuous time systems. Following a rather isolated chapter on entropy, we are then treated to a substantial account of stochastic perturbations for both discrete and continuous time. A number of different classes of perturbations are treated, overall giving a comprehensive account of the behaviour of noisy systems. Finally, a new chapter covers invariant measures, as opposed to densities, and concludes with a description of iterated function systems, popularised recently by Barnsley as a method of generating fractals and compressing images. There is no specific chapter dealing with applications, but rather a far wider variety of examples and physical and biological applications are scattered around the text.

Both books are well written, giving clear descriptions of the mathematical results that they treat, and of their proofs. Both include a wealth of examples, though those in Laws of Chaos tend to be more narrowly mathematical, whilst those in Chaos, Fractals, and Noise are more often motivated by physical or biological problems. The major difference in presentation between the two books thus lies in the area of motivation, and here Chaos, Fractals, and Noise really stands out. The authors have gone to a lot of trouble to attempt to explain important concepts, and the reasons for studying them, in an informal and non-technical manner. Indeed, the whole book begins which a chapter giving an intuitive description of densities and their importance to nonlinear dynamics. This makes the book ideally suited to the non-specialist reader, and I think almost everyone with an interest in dynamical systems would gain something from reading this book. By contrast, Laws of Chaos demands a higher level of mathematical sophistication from the reader and often gives him little support in figuring out why a particular result is being proved. I therefore suspect that it is far more appropriate as a reference book for those who are already familiar with the field.

Finally, I have to comment on the primary titles of both books, which as the description above should make clear, bear little relationship to their content. This is particularly so for Laws of Chaos, whose title suggests a comprehensive account of chaotic dynamics, but which concentrates entirely on a tiny subset of this field. The subtitles of both volumes are far more indicative of the material they cover and I suspect may have been the authors' original choices, with the primary titles imposed by publishers in a dubious effort to boost sales. This is a pity, since it only serves to detract from both books' otherwise excellent qualities.

UK Nonlinear News would like to thank Birkhäuser for providing a copy of Laws of Chaos for review.


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