UK Nonlinear News Book Reviews

Weighing the Odds: A Course in Probability and Statistics
D. Williams

Cambridge University Press 2001, 0-521-00618, 566 pages.£26 (paperback); £75 (hardback)ISBN: 0-521-80356

Reviewed by Jaroslav Stark

As those who have read some of my other book reviews in UK Nonlinear News will know, one of my key concerns about the state of the mathematical sciences today is the gulf that exists between applied mathematics and statistics. I find it astonishing that someone can get a 1st class degree in Applied Mathematics from almost any university in the UK (and most other countries) but when confronted by real data will have practically no idea how to compare it to a mathematical model. Conversely, modern statistics concentrates primarily on developing ever more sophisticated techniques for fitting and comparing models which from an applied mathematics perspective(and particularly from a nonlinear dynamics point of view) are rather simplistic. As a consequence, ask either an applied mathematician or a statistician how to fit data to a set of coupled nonlinear differential equations and, with a few notable exceptions, you are likely to be met by a puzzled shrug of the shoulders.I was thus intrigued to find a close parallel in the mischievous first sentence of the volume under review: Probability and Statistics used to be married; then they separated; then they got divorced; now they hardly ever see each other.The book?s stated aim is to provide support for a much-needed reconciliation. As a, perhaps unintended, by-product the book?s unusual approach and entertaining style results in an introduction to statistics which is unusually accessible to an applied mathematics audience. As an added bonus, the final chapter on quantum probability and computing will be of interest to anyone with a physics background. I doubt that there is another book anywhere that manages to combine topics as diverse as ANOVA and quantum entanglement in such an effortless way. Essentially no prior knowledge is assumed in either probability or statistics, resulting in a work that is suitable for a very wide range of backgrounds.Given the author?s background, the underlying emphasis is on concepts in probability, ranging from an intuitive first introduction to topics as advanced as martingales. This gives the volume a strong mathematical flavour, with analysis and linear algebra playing a key role. Statistical ideas are introduced gradually, and the author maintains a balance between the traditional frequentist approach and the Bayesian point of view. I suspect that he himself is more inclined philosophically towards the latter,but he is not above criticizing a pure Bayesian methodology where appropriate. One interesting aspect is his strong dislike of hypothesis testing, and preference for reporting confidence intervals of estimated parameters instead. This is a point of view that I increasingly sense in modern statistics, and it is useful to have it argued here so persuasively.A variety of other thought provoking and probably controversial ideas are sprinkled throughout the book, always presented in a lively and challenging manner. This makes for a very readable book, from which I learned a great deal and into which I have been continuously dipping since I finished it. I would recommend it to anyone, from final year undergraduate, to an established researcher in nonlinear science. They might be surprised how interesting statistics can be, and perhaps respond more positively next time they are asked to fit data to a model.

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

Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools
Didier Sornette

Springer, 2nd ed. 2004, 102 figs, 528pp., EUR 79.95, GBP 61.50, US $ 99.00Hardcover, ISBN 3-540-40754-5

Reviewed by Henrik Jense

This is an extremely impressive and comprehensive review of large areas of research into the collective behaviour of systems with many degrees of freedom. The author, Didier Sornette, is an exceptionally productive researcher who has contributed to most of the topics covered in the book.This doesn't by any means imply that the presentation is limited to Sornette's own work. On the contrary vast numbers of models and approaches are discussed in remarkable detail. The reference list contains a staggering1067 items. One gets the impression that Sornette knows these references in and out and presents the reader with a digested explanation of the essential content of this huge list of papers and books.The book does what the title promises, namely equip the reader with an arsenal of concepts and tools which will be a great help when reading the research literature or attacking research problems in the field of, what one might call, applied statistical mechanics. The focus is - again as the title implies - on the correlated behaviour of systems with many components. The term criticality is meant to imply that essential correlations exist between the different parts of the system and, hence, its behaviour cannot be deduced by a simple summation of the properties of the individual components.The tools and concepts supplied include a basic and worthwhile introduction to statistics. The first chapter introduces the very foundation of probability theory with a refreshing discussion of the frequency interpretation contrary to the Bayesian school. The material is organised in an unusual but very effective manner with an emphasis on characteristic functions, moments and cumulants.

I like very much the discussion of Extreme Value Statistics and Large Deviations in Chap. 1 and 3 sandwiching the obligatory discussion of the Central Limit Theorem in Chap. 2. The peculiarities of power law distributions (Levy Laws) are detailed in Chap.4, Fractals and Multifractals are explained with great lucidity in Chap. 5.Chap. 6 rounds off the mathematical tool box with a discussion of Rank-Ordering Statistics and Heavy Tails, topics which are central to the current activities in complex systems research.The remaining 11 Chapters are concerned with physics concepts and models. Abroad range of very relevant topics are covered: the relevance of the temperature concept to out-of-equilibrium systems, the role of long-range correlations and phase transitions. The latter is the prototype example of macroscopic coherent behaviour in physics; here the concept and its mathematical description are explained at a level that should be accessible to readers with no physics background. Two more methods chapters follow: one from dynamical systems theory on bifurcations and one from statistical mechanics on The Renormalisation Group.Chaps. 12 and 13 present two important models: The percolation model which is a paradigmatic model from statistical mechanics and the Rupture Model,which is an interesting model with a special appeal to materials scientists and geo-physicists.

Chap. 14 and 15 contains probably the material with widest current appeal and inspiration. Back in 1987 Bak, Tang and Wiesenfeld published a paper in which they introduced the concept of Self-Organized Criticality which they suggested to be the generic explanation of the power laws characterising many natural phenomena. This inspired a huge research effort which is still very active. Sornette devotes Chap. 14 to a detailed discussion of a large number of mechanisms that may lead to power laws and focus next on specific Self-Organized Criticality models in Chap. 15. I find these two chapters to be very useful nearly up-to-date summaries of the present research situation.The book's two last chapters contains for completeness discussions of random systems. Again well introduced and well explain material.This book is a treasure horn. Without assuming much mathematical background Sornette manages to supply the reader with the most essential mathematical tools and scientific concepts to be able to participate in the current research effort on developing mathematical modelling and understanding of extended system in which the behaviour is a result of cooperative interaction between the components. A very good book to have in the bag!This is the second edition and it is even by Springer, it therefore surprises me that there are small mistakes here and there. None of these are essential; a few might cause momentarily confusion for the reader unfamiliar with the material - I wonder if the copy editing has been sufficiently careful? Last point, do we use British or American spelling? Or both perhaps? I found on p. 96 line 9 and 11 a certain amount of oscillation between the two conventions. This is of course not important, but nor is it elegant.

Simulating, Analyzing, and Animating Dynamical Systems.
A Guide to XPPAUT for Researchers and Students
Bard Ermentrout

SIAM. Softback, 290 pages. ISBN: 0-89871-506-7.

Reviewed by Harvinder Sidh

Researchers in diverse areas such as Biology, Physics, Engineering etc are constantly investigating nonlinear phenomena in order to understand the behaviour of their specific physical system. Furthermore, nonlinear dynamics is now an integral part of many undergraduate Mathematics and Physics curricula in universities all over the world. Summer schools in areas such as ecology, physiology, medicine etc have also devoted much time to exploring nonlinear phenomena. Regardless of the application, the most important issue in the understanding of nonlinear behaviour is the combination of both analytical and computational techniques. However, a significant number of researchers and students from these diverse fields may have insufficient computational background to program in MATLAB, MAPLE,MATHEMATICA etc to solve the governing equations for the system that they are modelling. This is where XPPAUT is extremely useful. Bard Ermentrout,the author of this book and the developer of the software, states that XPPAUT offers several advantages over the above-mentioned softwares. These include:

a simple syntax for setting up various systems of equations such as ODEs, maps and some PDEs,
a convenient interface with AUTO - a powerful continuation and bifurcation package,
free downloading of the source code.

I must concur with Bard regarding the advantages of XPPAUT and see these as compelling reasons for anyone interested in studying nonlinear dynamics to use this software. Personally I feel that XPPAUT is an excellent software for both researchers and students. Prior to reviewing this book, I have always used MATLAB and AUTO in my research. However, I am now a ``convert''and have begun to use XPPAUT extensively in both my research and teaching ofdynamical systems. I must now stop talking about the software and proceed to review this book, although I find it very difficult to divorce one from the other.The book consists of nine chapters and seven appendices. The author is to be commended for his excellent organization of the subject matter and his clarity in writing. I found the background of the physical models used to illustrate particular aspects of XPPAUT to be extremely refreshing. Although contrived examples are often used in books, this is not the case here.The author begins with a useful explanation of how to install the package on computers with various operating systems. Chapter 2 explains the basics of creating an ODE file and solving the system and plotting the solutions. Due to the simplicity of the package, chapter 2 provides the reader with sufficient information to analyse their own system using XPPAUT. Chapter 3provides a more detailed explanation of the syntax of the ODE files. Here the author explains how to set up user-defined functions as well as the handling of discontinuous differential equations in XPPAUT. Chapter 4 whichis aptly entitled ``XPPAUT in the Classroom'' is one of the most useful chapters in the book. The chapter begins by showing how XPPAUT can be used to plot functions, before focussing on one-dimensional maps. Here the author shows how to obtain cobweb diagrams, bifurcation diagrams, Liapunov exponents, and the devil's staircase for such maps. The final part of this chapter is dedicated to nonlinear differential equations. The author explains very clearly how three- and higher-dimensional dynamical systems can be analysed using this package. Chapter 5 explains how to define ODE files for ``more complicated model systems'' such as systems with delay,integral equations, stochastic equations and differential algebraic equations. Chapter 6 focuses on the use of XPPAUT to solve boundary value problems and special cases of spatially distributed systems. I was very impressed with the ease with which the governing equations can be set up for such systems using XPPAUT. Chapter 7 is one of the major chapters as it clearly explains how to use XPPAUT's simple interface to the continuation package AUTO. Although I have previously used AUTO, I found the XPPAUT's interface much easier to use. Animating systems is the focus of chapter 8.Even I was able to produce good quality animations for some of my systems by following the author's clear step-by-step instructions and examples.Finally, chapter 9 is aptly entitled ``Tricks and Advanced Methods'' as it provides users with very useful tips including exporting data, linking with external C routines, and numerous other ways in which ``tricky'' systems or situations can be handled.Overall I believe that this is an excellent book for researchers and students who are interested in learning how to analyse dynamical systems using XPPAUT -- a powerful, simple-to-use and free software package. By using this book readers will be able to discover the software's full range of capabilities, and like me, they will certainly be impressed.

UK Nonlinear News would like to thank SIAM for providing a review copy of this book.

Nonlinear Dynamics in Physiology and Medicine
Ann Beuter, Leon Glass, Michael C. Mackey and Michèle S. Titcombe (editors),

Springer-Verlag, Interdisciplinary Applied Mathematics, Volume 25, 2003, 434pages, 162 illustrations, Hardcover.ISBN: 0-387-00449-1

Reviewed by Alona Ben-Tal

I was delighted when the book "Nonlinear Dynamics in Physiology and Medicine" arrived in the mail. This pleasant looking book is the most recent volume of the Interdisciplinary Applied Mathematics series by Springer-Verlag and since I know some other books in this series (for example, [1], [2], [3]) I looked forward to reading this new addition.The book has evolved out of notes written in three summer schools organized by The Centre for Nonlinear Dynamics in Physiology and Medicine at McGill University. The summer schools were held in 1996, 1997 and 2000. The result is an edited book with 12 contributors (mentioned later) that contains cutting edge subjects of research and (perhaps not surprisingly) reflects mainly the work done by the McGill group.Chapter 1, by M. C. Mackey and A. Beuter, gives "A wee bit of history to motivate things". The chapter contains some interesting examples showing the role mathematics has played in the life sciences and vice versa (but see also: highlights of a keynote address by Dr. Joel Cohen "Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better." http://www.bisti.nih.gov/mathregistration/ ).Chapter 2, by J. Bélair and L. Glass, "Introduction to Dynamics in Nonlinear Difference and Differential Equations", has nice examples of nonlinear phenomena seen in physiological systems, among them, the co-existence of two stable states in the human heart. Overall it is a reasonable introduction to the field but I was disappointed to find the following statement (given in p. 17 when the authors discuss the period three window seen in the logistic map):..."So why did Li and Yorke (1975) claim "period three implies chaos"? (Liand York 1975). The answer lies in the definition of chaos. For Li and Yorke, chaos meant an infinite number of cycles ....That definition does not involve the stability of the cycles."In the context of the book it is not clear that "claim" actually means"proved" [4]. I also thought that for the purposes of the book it would have been sufficient to give an intuitive definition of chaos, as the authors did on p. 16, skip the exact definitions of chaos altogether and refer the reader to a technical definition of chaos, given for example in [5, 6]. But isn't this an example where mathematics is a better microscope? We KNOW that an unstable chaotic solution exists even though we cannot observe it. I felt that a student reading this statement (and others on the definition of chaos) may get the wrong impression that only the observable solutions are important (and see for example, [7]).Chapter 3, by M. R. Guevara, is another introductory chapter to non-linear dynamics and contains some very nice examples of non-linear phenomena in physiology. Chapter 4, also by M. R. Guevara gives a nice introduction to the Hodgkin-Huxley equations and serves as a specific example to illustrate the subjects covered in Chapter 3.Chapter 5, by L. Glass looks at periodic solutions and the phenomena seen when an oscillating autonomous system is forced by a single stimulus or by periodic train of stimuli. A related subject, "Reentry in Excitable Media",is presented in Chapter 7 by M. Courtemanche and A. Vinet. Chapter 7describes the dynamics of excitable cardiac cell by three families of models: Cellular Automata, delay equations and partial differential equations.The important subject "Effects of Noise on Nonlinear Dynamics" is covered in Chapter 6 by A. Longtin. Different kinds of noise are discussed.Postponement of a Hopf bifurcation is demonstrated in a first order delay ed equation (model for a pupil light reflex) and the phenomenon of stochastic phase locking (usually known as "skipping") is described. The phenomenon of bursting as a result of noise is also presented but, to my disappointment,there is no discussion (or even mention) of bursting without noise.It seemed natural to me to read next Chapter 9 by J. Milton. This is a nice chapter that describes the pupil light reflex. I thought that there is a good balance here between physiology and mathematics but there is a relatively large number of typos.I next read Chapter 10 by A. Beuter, R. Edwards and M. S. Titcombe describing the interesting phenomenon of tremor (an approximately rhythmical movement of a body part). I thought it was interesting that the Van der Pol equation was proposed as a model for Parkinsonian tremor but there is hardly any discussion at all about this model. A six-unit Hopfield-type neural network model is discussed here in more detail.Finally, I made my way through the jargon of Chapter 8 (granulopoiesis,erythroid precursors, erythropoietin, neutrophil, am I reading English?,megakaryocytic, eosinophil, reticulocyte, idiopathic, promyelocytes, the list goes on, see http://cancerweb.ncl.ac.uk/omd/ for an On-Line Medical Dictionary). Chapter 8 by M. C. Mackey, C. Haurie and J. Bélair describes the control of blood cell production. Here you can find more examples of periodic behaviour on the scale of days that arise from Hopf bifurcations in delay equations.Throughout the book experimental results are presented and the authors share with the readers the joys and sorrows of dealing with real data including some practical aspects of estimating parameters (for example, a useful tip on obtaining data from published graphs by using Ghostview is given in Chapter 8 p. 248). All the chapters contain computer exercises with source codes and data files that are available on line athttp://www.cnd.mcgill.ca/books_nonlinear.html. The book also has three appendixes (an introduction to XPP, an introduction to Matlab and timeseries analysis) which I found handy. But the book lacks the coherence and general perspective that a single authored book has (despite the noted effort by the authors). Still, this is a stimulating book. I don't recommend it as a text book but certainly as a research and teaching resource. References:

J. Keener and J. Sneyd, "Mathematical Physiology", Springer-Verlag, 199
J. D. Murray, "Mathematical Biology I. An Introduction", Springer-Verlag, 3rd edition, 2002.
R. Seydel, "Practical Bifurcation and Stability Analysis. From Equilibrium to Chaos", Springer-Verlag, 2nd edition, 199
T-Y Li and J. A. Yorke, "Period Three Implies Chaos", The American Mathematical Monthly, Vol. (10), 197pp. 985-992.
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos", Texts in Applied Mathematics, Springer-Verlag, 2nd edition, 200
M. W. Hirsch, S. Smale and R. L. Devaney, "Differential Equations, Dynamical Systems & An Introduction to Chaos", Elsevier, 2nd edition, 200
C. Robert, K. T. Alligood, E. Ott and J. A. Yorke, "Explosions of Chaotic Sets", Physica D, Vol. 14 pp. 44-6
2000.

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