Nonlinear Dynamics and Chaos: Where do we go from here S.J. Hogan, A.R. Champneys, B. Krauskopf, M. di Bernardo, R.E. Wilson, H.M. Osinga and M.E. Homer (eds.) Institute of Physics Publ. (2002), price US$ 65.-, ISBN 07503808621.
Reviewer: Theo Tuwankotta, Institute of Technology (ITB), Bandung, Indonesia.

A yeshiva boy --- a young man studying in a rabbinical college --- took instruction from three rabbis. A friend asked him his reactions. "The first I found very difficult, disorganized, and poorly explained, but I understood what he was saying. The second was a lot clearer, and much more clever. I understood part of that." "And the third? They say he is very good." "Oh, he was brilliant! Such a magnificent, resonant voice --- it flowed as if from the heart. I was transported to realms beyond my imagining! So articulate, so lucid --- and I didn't understand a word."

Cited from the book: "The Collapse of Chaos" by Jack Cohen and Ian Stewart.

The quotation above reflects my impression after browsing through the book "Nonlinear Dynamics and Chaos: Where do we go from here", which is an extra-ordinary book. Exactly as indicated in the introduction by the editors, the theory of Nonlinear dynamics has become advanced. The applications penetrate into different fields of science and engineering. It is then a high-time to stop for a while and do some reflection, just as we always ask our students after doing a lot of exercises: What have we learned so far and what next.
The book is written by authors which are champions of their field. All researchers in Nonlinear dynamics should have access to this book. It is a valuable resource of references and it contains a lot of ideas and open problems in various field. One might think of it as a catalogue of problems in Nonlinear dynamics.
Most authors have presented their share in the book in a very inspiring way: presenting ideas and more importantly, the reason behind those ideas. Reading chapter by chapter of the book, one might even be able to feel the excitement of the symposium hosted at the University of Bristol.

The introduction of the book is a "must-read". It presents the nature and the philosophy of the book (and the symposium). Reading the introduction, the editors clearly have done a great job of managing each of the invited lecturer to translate the philosophy of the symposium into their lectures.

The book covers three streamlines: neural and biological systems, spatially extended systems, and experimentation in the physical sciences. As noted in the introduction, it is surprising to see that many chapters of the book address two or even all of the three topics. I will discuss some chapters which are appealing to my own interest.

The book starts with a chapter by John Guckenheimer on Bifurcation and degenerate decomposition in multiple time scale dynamical systems. Multiple time scale dynamics has been a hot topic of research for a long time. Applications of such interesting dynamics are abundant, varying from chemical reactions to lasers dynamics. The complexity of the dynamics due to the existence of various time scales combined with higher spatial dimension is more or less the message in this chapter.

Chapter two is by Robert Mc. Kay and the title is Many-body quantum mechanics. The author proposed this topic as one of the promising lines in research on nonlinear dynamics. This chapter provides the reader with various open questions ranging from fundamental mathematical questions such as: Can we develop a theory of dynamical systems on a non-commutative manifold up to quite technical questions such as developing a quantum analog of, for instance, discrete breathers and homoclinic chaos. There is also a section on a possible solution to some of these questions. The author clearly has interaction between applications and theory in mind.

The next chapter is written by Uwe an der Heiden with the title Unfolding complexity: hereditary dynamical systems --- new bifurcation schemes and high dimensional chaos. The author explores dynamical systems which have memory. One of the exciting sections in this chapter is on second-order non-smooth difference-differential equations. Non-smooth systems are well-known for their complexity and nontrivial dynamics. The technique that is used to analyze such a system is also quite different from those in smooth systems. The section also reflects the difficulties in finding harmony between smooth and non-smooth systems.

An important application of the theory of dynamical systems is found in the theory of control. A question like how to create stability out of unstable state is a topic which is covered by Christopher K.R.T. Jones in the fourth chapter Creating stability out of instability. The author deals with problems arising in nonlinear optics. Two approaches are described, one involving switching in time and the other using spatial distribution of the unstable phases. Both, however, lead to stabilization.
The author poses many interesting fundamental questions. The first one deals with linear theory versus nonlinear theory (as well as the local versus global theory of dynamical systems). Is it essential to introduce nonlinearity or is it actually destroying the control?

Pattern formation is a topic which arises in many applications. Chapter six, which is written by Edgar Knobloch, deals with this subject. It is amazing to note that completely different systems behave in an almost identical fashion. Related to the presentation in chapter four, the author of this chapter also addresses the question of local versus global analysis. The author points out interesting aspects of weakly nonlinear systems (which are obtained most of the time by localizing around a particular steady state) as compared to the fully nonlinear system. It is a very nice survey paper which points out interesting open problems and contains a nice source of references.

One of the first systems known to exhibit chaos is the Lorenz system. This system is actually a caricature of the flow in the atmosphere in the sense that the Lorenz system is three dimensional while the original flows are infinite dimensional. It is natural then to expect that turbulence which is a property of fluid motion might be explainable by chaos. This is the theme of chapter seven titled Is chaos relevant to fluid mechanics. The author, Tom Mullin, starts with the fact that although there are lots of things already known about chaos, it seems that the problem of turbulence remains a mystery. Taylor-Couette flow is chosen to illustrate this. However, the author states his believe that low-dimensional dynamics might be able to provide the basic ingredients to explain turbulence.

There are six chapters that I did not mention in this review due to my own personal taste in research. However, my impression is that all authors did a good job presenting the excitement of their research and addressing the interesting questions. This book in general is a valuable addition to the literature of the theory and practice of nonlinear dynamics and chaos.