UK Nonlinear News , August 1997.

Nonlinear Dynamics

H.G. Solari, M.A. Natiello and G.B. Mindlin

Institute of Physics Publishing, Bristol, UK

Reviewed by Jerry Huke

The rise in popularity of dynamical systems theory, fuelled largely by a widespread interest in the phenomenon of chaos, has led to an increasing flood of textbooks on the subject onto the market, and it would appear that any new one would face stiff competition. However, the book under review occupies a niche that is not well supplied; it is a graduate level book, aimed specifically at physicists who want to apply the tools of nonlinear dynamics to their own research problems. As such it sets itself the ambitious targets of not only describing nonlinear dynamics, but of presenting the main mathematical techniques and of showing how these are applied in the investigation of experimental and observational data. Since any one of these in itself could (and currently does) form the subject matter of a book, it is not to be expected that the authors can provide a comprehensive and detailed coverage. What they do provide is a stimulating and challenging book that conveys well the flavour of nonlinear dynamics as an active area in physics, and that hopefully will encourage physicists to acquire the ideas and apply them in a broad range of contexts.

The style of the book is rather different to that of the more well established books at this level, (notably J. Guckenheimer and P. Holmes's "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields"): the authors go to great lengths to set the theory in the context of physical problems, describing how the problems engender and shape the theories and the associated mathematics. This style of presentation leads them, for example, to introduce homoclinic bifurcations in the context of numerical experiments on the Lorenz system, to follow with a general discussion of global bifurcations in which homoclinic tangles and horseshoes are introduced, and then to give a more detailed discussion of horseshoes and their symbolic dynamics. A more mathematically oriented presentation would probably have taken these subjects in just the reverse order. Other manifestations of the authors' theme of nonlinear dynamics as a tool for physicists are: the inclusion of a chapter on numerical experiments, illustrating how such simulations are an important source of information about models, prompting and guiding the subsequent theoretical analysis; and the inclusion of a chapter on how experimental data can be manipulated to investigate the dynamics of the underlying system, using embedding techniques.

Most of the usual topics in nonlinear dynamics get an airing: there is a general introductory chapter describing flows, orbits, attractors, Poincaré sections and so on, and there are chapters on stability, local bifurcations, global bifurcations, maps of the interval, and averaging methods. Some less common topics also make an appearance: there is a chapter on the role of symmetry in bifurcations, and one on the topological properties of three dimensional flows.

With this much ground to cover it is inevitable that the authors spread themselves rather thinly at times. The discussion of some of the mathematical tools (such as local Poincaré maps, or Liapunov exponents) is sketchy, and could leave the uninitiated reader unsure about how these things are actually used. Since this book is aimed primarily at physicists and other physical scientists there is very little in the way of formal proof; the more intuitive arguments which are presented instead will appeal to many, but may not be to everyone's taste, especially not those with a more mathematical bent. Part of the answer to this lies in the remark in the Preface that "this is a book to be read with paper and pencil at hand". This is certainly true: the reader is often left to fill in the details for him/herself, and will no doubt get the most out of the book by doing so. But as well as paper and pencil, occasional reference to Guckenheimer and Holmes might be useful to flesh out some of the arguments.

Apart from a willingness to put pencil to paper, the book requires a grounding in elementary topology, (open and closed sets, and homeomorphisms are talked about with little preamble). For some physics graduates this could be a problem, but a standard ancillary analysis course, up to and including metric spaces, would probably provide sufficient background. Some familiarity with ordinary differential equations would be helpful, as would some knowledge of linear systems.

Though it makes fairly serious demands on the reader, the research physicist who works through this book will gain rich rewards in a broad familiarity with nonlinear dynamics and its applications. It might even cause them to look at the phenomena they study in a new way.

Jerry Huke, Mathematics Department, University of Manchester Institute of Science and Technology.

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