What led us to write this book? A fair question. Is the world clamoring for a lucid introduction to nonlinear chemical dynamics? Did our wives and children implore us to share our insights into this arcane subject with an eager but ignorant world?

We would be the last to suggest that the subject matter treated in this volume is the stuff that bestsellers are made of. These topics are, nonetheless, of interest to a growing number of scientists and engineers, not only chemists, but physicists, biologists and others in a variety of obviously and not so obviously related fields. Three decades ago, a book devoted largely to chemical oscillations would have been inconceivable. Most chemists then viewed such behavior as a form of perpetual motion, rendered impossible by the Second Law of Thermodynamics. Fifteen years ago, one might have imagined writing a book of this sort, but it would have been a thin work indeed, since only two chemical oscillators were known, both discovered by serendipity, and only one understood to any extent at a mechanistic level.

Times change, and today chemical oscillations and the more encompassing field of, nonlinear chemical dynamics are among the most rapidly growing areas of chemical research. Both of us, teaching at very different universities, have observed that disproportionate numbers of graduate and undergraduate students flock to do research in this area. The visually compelling phenomena and their potential significance to a wide range of problems make nonlinear chemical dynamics a subject about which colleagues in such disparate fields as neurobiology, polymer science and combustion engineering seek to become better informed.

One of the greatest handicaps facing potential entrants into, or consumers of the fruits of this field of research is the lack of an introductory text. This gap cannot be accounted for by the inherent difficulty of the subject. The mathematical and chemical tools required are almost all possessed by any well trained undergraduate in the sciences. Most of the necessary experimental apparatus is, by modern standards, nearly primitive and certainly inexpensive. This last feature accounts in large measure for the many significant contributions that have been made to this field by scientists from Eastern Europe and the developing world.

There are, to be sure, some excellent books of which the reader should be aware and to which we owe a real debt. Limitations of space, both on this page and in our immediately accessible memory banks, confine us to mentioning but a few here. More will be found in the bibliography. Nicolis and Prigogine [Nicolis and Prigogine, 1977] discuss the foundations of nonequilibrium thermodynamics, laying to rest the misguided notion that chemical oscillation and its cousin, dissipative structures, violate some physical principle or other. They also show, largely by using the classic Brusselator model, that periodic oscillation and spatial wave propagation can arise from very simple mathematical models. They do not, however, at a time when experimental observations of nonlinear dynamical phenomena in chemistry were extremely limited, devote much attention to experimental aspects. Field and Burger [Field and Burger, 1985] edited a comprehensive collection of essays, nearly all of which focus on the behavior of the archetypal Belousov-Zhabotinsky reaction. Recently, Gray and Scott [Gray and Scott, 1990] and then Scott [Scott, 1991] have produced lucid treatments first of simple models of nonlinear oscillations and waves in chemical systems and then of chemical chaos. Much of the most exciting work in this rapidly moving field has appeared in collections of conference papers, beginning with the results of the 1968 Prague conference on biological and biochemical oscillators [Chance, et al., 1973] that may be said to have launched nonlinear chemical dynamics as a serious field of inquiry.

None of these volumes, however, is satisfactory in our view as a text for a course at the advanced undergraduate or introductory graduate level or as a means for a newcomer to the field to obtain an overview and a relatively painless means of access to a basic competence in his or her area of interest. We believe strongly that the subject can be taught and learned at this level!

It is always tempting in teaching a course or writing a book to focus on theory; it lends itself more easily to the blackboard or to the printed page. Chemistry, though, and nonlinear chemical dynamics in particular, is an experimental science. When chemical oscillations existed primarily in Lotka's models [Lotka, 1925], there was no subject of nonlinear chemical dynamics. When Turing structures could be found only in the papers of mathematical biologists, they played only a tiny role in this field. We have tried in this book to convey the experimental as well as the theoretical background of the subject. We describe how to build a flow reactor, for example. We provide appendices that contain recipes for lecture demonstrations and guidelines for laboratory experiments. We recommend that the reader try at least some of the demonstrations. They are just the sort of thing that hooked many chemists at an early age; solutions suddenly switch from one color to another -- not just once, but repeatedly. The demonstrations have provoked gaping mouths and perceptive questions from audiences ranging from elementary school children to university boards of trustees.

We have chosen to divide the book into two sections. In the first, we present an overview of the subject. We start with a brief history and then move on to review some of the basic mathematics and chemistry. We next discuss the flow reactor or CSTR, an experimental tool borrowed from chemical engineers, which led to the rapid expansion of nonlinear chemical dynamics in the 1970's and 1980's. The CSTR allows one to design new chemical oscillators, avoiding the earlier procedure of stumbling upon them. Having outlined how to build a chemical oscillator, we proceed to look at the task of dissecting them, i.e., of constructing molecular level descriptions or mechanisms. A realistic view of most chemical systems takes into account their behavior in space as well as in time. In the systems of interest here, this means that one must consider diffusion and how it can lead to pattern formation and wave propagation, a subject we consider in Chapter 6. We close our overview with a brief discussion of some of the computational tools that have furthered the understanding of these complex chemical systems.

A one-semester course in nonlinear chemical dynamics or an introduction for someone intent on entering the field as a researcher might consist of Part I (some or even much of which will be review, depending on the reader's background) supplemented by one or two of the chapters in Part II. Some of these "special" topics are treated in other books, but others (e.g., delays, polymers, convection) are treated in a context that is both chemical and pedagogical for the first time here. Each chapter in Part II can be read independently of the others, though readers may (we hope will) find that there will be symbiotic effects among certain combinations of chapters (e.g., 9 and 15 or 11, 12 and 13), some of which may not yet have occurred to the less than omniscient authors. A reasonable year's course in this subject could cover all of the special topics plus selected papers from the current literature and/or projects proposed by the instructor and the students. We have found that students are eager to get in there and do something on their own and that it is not unrealistic for them to do so. Indeed, some of the earliest and most significant discoveries in both of our laboratories were made by undergraduates.

Both of us "wandered" into this field after being trained in other areas. Having seen many others come this way and never look back, we are convinced that the direction we took is a natural one. We hope that this book will make the road just a little smoother for those who follow us. Remember, if you have any doubts, try the demonstrations!



September 1997