Dynamic pursuits that led to chaos

According the mathematician John von Neumann, "the origin of mathematics lies in empirics". There is no better illustration of this general dictum than the movement of the planets, which served to motivate much of the mathematics of the ancients.

This book is a continuation of that venerable story into modern times, providing an exposition of the manner in which problems of planetary motion have led to what today is termed "dynamical system theory". About the most natural question one can ask about the movement of the planets is whether at some point in the future there will be a collision of two (or more) planets or whether some planet will acquire a velocity so great that it escapes from the solar system? This, the so-called N-Body problem, is certainly the most famous question in dynamical system theory. And the reason it is famous is that no one knows the answer for a system containing more than one planet.

In 1885, King Oscar II of Sweden offered a prize for the solution of this problem. It was awarded in 1889 to the famed French mathematician Henri Poincare. The judges felt that Poincare's approach and methods were so novel and shed so much light on the problem that he merited King Oscar's prize, even though he did not solve the problem of the stability of the solar system completely.

The story recounted in this book is how the systematic pursuit by several generations of mathematicians of Poincare's geometrical concepts have led to the modern field of dynamical system theory, culminating recently in the highly publicised work on "chaos". The authors have done a first-rate job of historical exposition, providing an overview of the idea of a dynamical system as it emerged from Poincare's prize paper. Moving from this now-classic work, the book recounts the major advance by Berkeley mathematician Steven Smale in the 1960s.He invented a dynamical system called a "horseshoe map", which served as a concrete example of just how crazy the behaviour of a dynamical system could be.

From Smale's work, the authors go on to tell the story of the furious development of these notions by the Russians A. N. Kolmogorov and V. I. Arnold, as well as the German-American Juergen Moser, with certain types of dynamical systems that serve as a kind of forerunner to full-fledged chaotic systems.

Strangely, perhaps, in a book on modern dynamical system theory, the authors give nothing but passing discussion to chaos. But given the plethora of volumes at all levels that have been written in the past decade on chaos, they are to be commended for using such good judgement in omitting this over-exposed topic. Stability theory and planetary motion are quite enough.

Who is the audience for this book? It is much too technical for the general reader, even the proverbial "educated layman". On the other hand, it is not really for the mathematically trained either, at least those whose training involved even a small dose of dynamical system theory or differential equations. They will have seen much of the book's technical material before.

This confusion over the book's target audience also shows up in the style of exposition at various spots, when the authors fall into a kind of fictional style. For example, the book's first paragraph talks about Poincare pushing his chair back and going out for walk on a beautiful spring afternoon. This kind of opening statement is completely normal narration - in a novel. But it is totally out of place in a work of non-fiction, especially one at this level of technical detail. In my view, these are the passages that should have been starred, not the ones the authors felt were too technical. I can only conclude that the real target reader for this book is someone who has both the mathematical training not to be intimidated by the jargon and technical diagrams, and a strong desire to know about the history of dynamical system theory as it has unfolded from Poincare's time up to the 1970s. I am not sure how large this group is, but those rare folks possessing these twin traits will get far more than their money's worth from this book. For them, it is a gem.

John Casti is resident member, Santa Fe Institute, New Mexico, United States.