UK Nonlinear News, August 1999


Random Dynamical Systems

By Ludwig Arnold

Reviewed by Jaroslav Stark

 

Springer-Verlag 1998, ISBN 3-540-63758-3, 40 figures, 586 pages, £57.50.

There is only one possible word that can be used to describe this book: seminal. It is perhaps difficult to remember today, when every bookshop has a plethora of good nonlinear dynamics titles available, the dearth of textbooks in this area even just 15-20 years ago. This situation was significantly altered by the publication of Guckenheimer and Holmes’ Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. This was arguably the first book that brought together the pure and applied sides of the subject and presented them in a way that was comprehensible to scientists and engineers in the applied sciences. It had a profound effect on the whole area of applied nonlinear dynamics and on a whole generation of PhD students, myself included. It was undoubtedly one of the main stimuli for the tremendous growth of the subject in subsequent years, and even today remains one of the best textbooks in the field.

I believe that Ludwig Arnold’s book on Random Dynamical Systems will play just as important a role in the coming decades. The author has set himself the task of writing a "continuation" of Guckenheimer and Holmes into the area of stochastic dynamics. In this he has succeeded admirably, and in my opinion this volume will set the agenda as nonlinear dynamics tries to come to terms with the fact that most real systems are subject to significant levels of noise. The importance of this field cannot be overestimated. In the century or so since Poincaré and Liapunov we have made great strides in understanding dynamical systems. Most of this effort has been focused on the deterministic case. Whilst there are certainly still many deep issues to be resolved in this area it does seem to me that the dynamics community is in danger of concentrating more and more on unravelling ever more intricate phenomena under ever more restrictive hypotheses. At the same time, only a handful of people are working on the fundamental problems that arise when dynamical systems are subject to random effects. Of these, the group built up by Ludwig Arnold in Bremen is one of the best in the world and has made a major contribution to the development of the subject. His former research students and research assistants have been instrumental in spreading this work throughout the world, and his book will surely accelerate this process.

It is a wonderful volume. Hitherto progress in this area was hampered by the different viewpoints of the dynamics and stochastic communities. This volume manages a seamless marriage of the two, and there is a great deal that will be familiar to those working in deterministic dynamics: fixed and periodic points, attractors, linear stability analysis, invariant manifolds, bifurcations and even shift spaces. At the same time there is a comprehensive account of the fundamentals of stochastic differential equations. All of this, and more, is presented in a beautifully clear and intuitive way. Those who have heard the author lecture will know that he is one of the best speakers in the field of dynamics today, and the presentation in this book lives up to his usual high standards. What is particularly helpful is the way each new concept is given an informal and intuitive description before receiving a full rigorous treatment. One thus always has a clear idea of what one is trying to achieve even when delving into the midst of complex technical mathematics.

Having said this, it has to be emphasized that most will not find this an easy book to read. This is not through any fault of the author’s, but is rather a function of the difficulty of the subject, and perhaps of the lack of suitable mathematical background for the vast majority of us working in deterministic dynamics. This will particularly be the case for graduate students. I doubt that there will be many starting PhD’s in the UK who will have sufficient preparation from their undergraduate degrees to appreciate this book. This I think is a sad indictment of what we teach our students, and perhaps of the overall direction in which nonlinear dynamics is heading, as remarked above. Hopefully, the appearance of this book will help to address this imbalance, and I personally envisage that in ten years time this will be one of the two or three standard books that I will give to my new research students to read.

I would like to end on a personal note. Those who know Ludwig Arnold will be aware of the immense effort that he has put into writing this book over the last decade. Many gaps in the material had to be filled and "folklore theorems" had to be proved. A great deal of thought went into how best to present difficult and often subtle questions and a tremendous amount of work was required to create a unified interdisciplinary account. The end result is clearly worth it, and a great service to the community. Thank you.

Dr. Jaroslav Stark,
Tuesday, July 27, 1999.

 

UK Nonlinear News thanks Springer-Verlag for providing a copy of this book for review.


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