Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems

J.-R. Chazottes and B. Fernandez, eds.

Lecture Notes in Physics 671, Springer (2005), 361 pp.

I was given this book to review shortly after submitting my PhD thesis, and over two months before I was due to start my new job. However, having a lot of free time quickly mutated into being extremely busy, and I found myself carrying the book with me everywhere, snatching time to read it whenever I could. Luckily, the structure of the book lends itself nicely to these situations, as each of the chapters is self-contained, and none of them are overly long, although in general, the material is at quite a high level, and requires much thought from the reader.

The book is made up of lecture notes from the CML2004 meeting in Paris on the theory of coupled map lattices (CML) and related topics. The contributing authors make quite an impressive list of well-known figures in the field. The editors, Jean-René Chazottes and Bastien Fernandez give a very good introductory chapter, with a simple explanatory description of a CML and a good overview of the book's contents. A CML is a discrete time dynamical system which is extended in space over a discrete lattice or network. The dynamics at each site in the spatial lattice is governed by a local map, and usually there are couplings between the dynamics at each of the sites. I particularly liked the diagram on page 3 which schematically represents the different dynamical regimes of CML, as distinguished by the strength of the coupling, and the type of dynamics given by the local map.

The book is divided into four sections, described in the introduction. On reading the introduction, it was not immediately clear to me where each of these sections started and ended, and the book would probably have benefited from a higher level of structure above that of `chapter'. Having said this, I suspect one of the reasons preventing this may be that the edges between the sections are not particularly well defined. The first section of the book looks are statistical properties of coupled chaotic maps, and the introductory chapter hints at links with statistical mechanics. Section 2 describes geometric aspects of lattice dynamical systems, section 3, spatially extended systems with monotone dynamics and section 4 gives some specific examples of CML in the biological sciences.

The chapter by Leonid Bunimovich gives a very good introduction to the theory of CML which requires very little prior knowledge at the start. He makes good use of specific examples to emphasise the ideas and demonstrate the theorems. Towards the end of the chapter, more use is made of fairly advanced measure theory, this is indeed true of much of the book. I was slightly disappointed to notice a number of editing errors in this chapter, although this does not seem to be consistent throughout the whole book.

The remaining chapters in the first section were at a more advanced level. I was pleased to find that the chapter by Wolfram Just and Frank Schmüser expanded on the links between CML and Statistical Mechanics which were hinted at in the introduction. The middle two sections of the book contain a lot of new information, but it is generally well-written, and good use is often made of diagram and schematic pictures.

The last three chapters make up section 4 of the book and contain specific examples of CML. Yuri Maistrenko, Oleksandr Popovych and Peter Tass discuss the well-known Kuramoto model, applying it to various biological situations, including deep brain stimulation. Hidde de Jong and Ricardo Lima apply CML to gene regulatory networks and Bard Ermentrout discusses networks of coupled neurons. I found these chapters fun to read as the level of mathematics was significantly lower than in the rest of the book and it was interesting for a mathematician to learn some biology.

Overall, I found this an enjoyable and interesting book, but for someone who had no previous research experience in CML, it was pretty tough going in some places. I would recommend it to the dedicated post-graduate or researcher willing to put in some time to learn about an interesting subject.

Reviewed by Claire Postlethwaite

UK Nonlinear News would like to thank Springer for providing a copy of this volume for review.