Bifurcation for beginners

Introduction to Nonlinear Science - Stability, Instability and Chaos
September 29, 1995

One of the problems facing university science departments as they see their pool of prospective new undergraduates ebbing away year after year is how to bridge the gulf between general and technical description. The study of chaos and complexity has been widely popularised in the media and many books have been devoted to explaining its key insights and central paradigms. Such accounts lead students of physics and mathematics to expect that they will learn more about these fascinating subjects from day one at university. Nothing could be further from the truth. A similar anticlimactic pattern characterises the study of particle physics or cosmology. A considerable amount of basic education is required before any of these attractive subjects can be tackled at the level of a participant rather than a mere spectator. The time needed to cover this foundational material along with the necessary mathematical techniques leaves little space for the inclusion of courses on developing subjects until the final year of the course when advanced options are traditionally chosen.

These two books are advanced teaching texts devoted to some aspects of chaotic behaviour that have been widely reported at a general level in the press. They focus upon the stability properties of ordinary differential equations and difference equations; both have emerged out of university courses the authors have taught. Paul Glendinning's book has grown out of recurrent final-year undergraduate courses in nonlinear differential equations and bifurcation theory at Cambridge, while G. Nicolis's book derives from the lecture notes for a 30-hour course about nonlinearity for scientists with a mathematical background. Both books contain problems (but no solutions) and are informal in the sense that the proofs and mathematical descriptions are designed to be the clearest and most readily intelligible to the reader rather than the quickest and most elegant. But the two books are quite distinct in style and consequently in their likely (if not intended) audiences. Glendinning provides material for the sort of final-year course that would be taught as an option in applied mathematics. The author tries to avoid too much formality and pure mathematical rigour but there is little physical motivation for the mathematical systems that are studied and this would probably make this book difficult to use in theoretical physics courses. However, for undergraduates in mathematics the book will be attractive. The author writes clearly and carefully, weaving together general results with a steady supply of simple examples and exercises for the reader. Pick a section of the book to read at random and one is immediately confident that effort is going to be rewarded; that the author is really explaining rather than simply recounting. He says that in structuring the course he had an eye towards subjects that are well suited to the setting of problems and the exercises are very carefully chosen, having emerged over a long period of teaching this material to successive audiences. These audiences, none the less, were clearly of above-average ability and anyone using this book to teach a series of courses on nonlinear systems will have to be selective about some of the more challenging material.

Despite covering much of the same stuff of linear stability theory, phase plane analysis of ordinary differential equations, bifurcation theory and the chaotic dynamics of one-dimensional maps, Nicolis's book has a quite different feel to it. It is full of analyses of particular physical problems to motivate the study of nonlinear systems and this reflects his intended audience: chemists and physicists with interests in nonlinearity. There is detailed discussion of the Benard convection problem, the Landau-Ginzburg equation, spatial pattern formation, and a variety of chemical oscillators. Many of his detailed examples reflect his research interests and those of the Brussels group. The discussion is at a high level throughout but without the same amount of explanatory detail supplied by Glendinning. The flavour is that of an advanced graduate course. I do not believe that this book would be suitable for undergraduate courses in British universities. Certainly the audience of mathematically equipped scientists he addresses does not exist at that level in this country. But at graduate level this book could form the basis of a course that could simultaneously serve the needs of mathematicians, physicists, and theoretical chemists. Unfortunately, neither of these books will be very helpful to biologists. The mathematical level is too high even for graduate courses and the examples used seem to avoid biological problems completely; nor does either book discuss topics like self-organising criticality.

Nicolis's style is that of the physicist, picking instructive particular examples to illustrate classes of interesting nonlinear phenomena, but the development of the subject proceeds rapidly and there is much detail to fill in which Glendinning would have provided. The problems involve calculation but often conclude by asking a penetrating question. The absence of solutions is therefore unfortunate. Whereas Glendinning contents himself with expounding the core of the subject, leaving glimpses at the frontiers to the guide to further reading, Nicolis is continually making reference to research papers and unsolved problems. For the research student seeking to work in some area of nonlinear science this will be stimulating. But if I apply the randomly selected passage test to Nicolis's book then the text gives the impression of being a little too fast moving, assumes the reader knows many other things, and frequently uses advanced technical terminology. As a result one does not get a confident feeling that what is written on a page of this book is all one needs to understand it, as one does with Glendinning.

None the less, for the specialist researcher Nicolis is usually more interesting, sometimes speculative, always oscillating between being a teaching text and a research monograph: the sort of book that one uses to upgrade your basic knowledge of a subject or to relate it to other fields. Glendinning is solid and dependable. The sort of book to which you turn to learn something for the first time, confident that he is guiding one along a path that many others have successfully followed. These are both novel books of great value, carefully written by researchers who are experts in particular branches of nonlinear science and experienced in expounding its mysteries to advanced students.

John D. Barrow is professor of astronomy, University of Sussex.

Introduction to Nonlinear Science

Author - G. Nicolis
ISBN - 0 521 46228 2 and 46782 9
Publisher - Cambridge University Press
Price - £35.00 and £13.95
Pages - 254

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