Subject: ARTICLES - Part (1/5) of UK Nonlinear News
From: UK Nonlinear News (uk-nonl_at_ucl.ac.uk)
Date: 09/01/04
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Date: Wed, 01 Sep 2004 15:27:45 +0100
Subject: ARTICLES - Part (1/5) of UK Nonlinear News
UK Nonlinear News, August 2004
------------------------------------------------------------------------
Articles and Reviews
* Book Review: Simulating, Analyzing, and Animating Dynamical
Systems. A
Guide to XPPAUT for Researchers and Students .
Reviewed by Harvinder Sidhu.
* Book Review: Critical Phenomena in Natural Sciences. Chaos,
Fractals,
Selforganization and Disorder: Concepts and Tools .
Reviewed by Henrik Jensen.
* Boor Review: Weighing the Odds: A Course in Probability and
Statistics
Reviewed by Jaroslav Stark
* Book Review: Nonlinear Dynamics in Physiology and Medicine
Reviewed by Alona Ben-Tal
* Report: Noisy Oscillators - Workshop on the Dynamics of Coupled
Oscillatory & Complex Systems
By Peter V.E. McClintock
* Report: Modelling of Neuronal Dendritic Trees
By Yulia Timofeeva
* A listing of reviews of nonlinear books can be found at
http://www.amsta.leeds.ac.uk/Applied/news.dir/issue7.dir/art/books.html
(this article is periodically updated).
* An index of UK Nonlinear News can be found at
http://www.amsta.leeds.ac.uk/Applied/news.dir/uknonl-index.html.
------------------------------------------------------------------------
Simulating, Analyzing, and Animating Dynamical Systems. A Guide to
XPPAUT
for Researchers and Students
Bard Ermentrout
Reviewed by Harvinder Sidhu
SIAM. Softback, 290 pages.
ISBN: 0-89871-506-7.
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 of
dynamical 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
3
provides 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
which
is 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.
------------------------------------------------------------------------
Critical Phenomena in Natural Sciences. Chaos, Fractals,
Selforganization
and Disorder: Concepts and Tools
By Didier Sornette.
Reviewed by Henrik Jensen
Springer, 2nd ed. 2004, 102 figs, 528pp., EUR 79.95, GBP 61.50, US $
99.00
Hardcover, ISBN 3-540-40754-5
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
staggering
1067 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. A
broad 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.
------------------------------------------------------------------------
Weighing the Odds: A Course in Probability and Statistics
D. Williams
Reviewed by Jaroslav Stark
Cambridge University Press 2001, 0-521-00618, 566 pages.
£26 (paperback); £75 (hardback)
ISBN: 0-521-80356
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.
------------------------------------------------------------------------
Nonlinear Dynamics in Physiology and Medicine
Ann Beuter, Leon Glass, Michael C. Mackey and Michèle S. Titcombe
(editors),
Reviewed by Alona Ben-Tal
Springer-Verlag, Interdisciplinary Applied Mathematics, Volume 25, 2003,
434
pages, 162 illustrations, Hardcover.
ISBN: 0-387-00449-1
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"?
(Li
and 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 7
describes 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
delayed
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 at
http://www.cnd.mcgill.ca/books_nonlinear.html. The book also has three
appendixes (an introduction to XPP, an introduction to Matlab and time
series 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.
UK Nonlinear News would like to thank Springer-Verlag for providing a
copy
of this volume for review.
References:
1. J. Keener and J. Sneyd, "Mathematical Physiology", Springer-Verlag,
1998.
2. J. D. Murray, "Mathematical Biology I. An Introduction",
Springer-Verlag, 3rd edition, 2002.
3. R. Seydel, "Practical Bifurcation and Stability Analysis. From
Equilibrium to Chaos", Springer-Verlag, 2nd edition, 1994.
4. T-Y Li and J. A. Yorke, "Period Three Implies Chaos", The American
Mathematical Monthly, Vol. 82 (10), 1975, pp. 985-992.
5. S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems
and
Chaos", Texts in Applied Mathematics, Springer-Verlag, 2nd edition,
2003.
6. M. W. Hirsch, S. Smale and R. L. Devaney, "Differential Equations,
Dynamical Systems & An Introduction to Chaos", Elsevier, 2nd
edition,
2004.
7. C. Robert, K. T. Alligood, E. Ott and J. A. Yorke, "Explosions of
Chaotic Sets", Physica D, Vol. 144, pp. 44-61, 2000.
------------------------------------------------------------------------
Noisy Oscillators: Workshop on the Dynamics of Coupled Oscillatory &
Complex
Systems
Ljubljana, 10-13 December 2003
By Peter V.E. McClintock
The Workshop on the Dynamics of Coupled Oscillatory and Complex Systems
was
held at the University of Ljubljana, Slovenia, 10-13 December 2003. The
venue was the Zborni?na dvorana in the historic central building of the
University in the middle of the city, a stone's-throw from the
Philharmonie
where Mahler worked for many years (and Beethoven's job application was
rejected!).
This was a small select gathering, mostly by invitation, and highly
interdisciplinary. The 37 participants, drawn from 17 countries,
included
all ages and stages from selected PhD students up to senior scientists.
The
meeting was opened by Professor Katja Breskvar, Vice-Rector of the
University of Ljubljana. Professor Toma? Slivnik, Dean of its Faculty of
Electrical Engineering, contributed some further welcoming remarks and
the
first scientific session then followed immediately.
The meeting reflected the fact that, in the modern theory of complex
dynamical systems in physics (fluctuations in lasers), biology
(transport of
proteins), physiology (autonomous nervous control of the cardiovascular
dynamics), and econometrics the effects of both chaotic and stochastic
dynamics, and their mutual interactions, must be taken fully into
account.
It was also motivated by a growing awareness that reaching an
understanding
of the general features of chaos-noise interactions in complex dynamical
systems may introduce entirely new ideas of how to control the system
dynamics, and of how to exploit its complexity usefully in a variety of
applications on all scales from molecules to the global economy.
Examples
discussed were numerous, including e.g. applications of stochastic
resonance
and Brownian ratchets in biology, chaotic communications, and chaos
control.
All these systems operate in regimes far away from thermal equilibrium,
and
the problems of modelling are highly challenging.
The function of the meeting was thus to bring together fluctuational
dynamicists working on nonequilibrium and complex systems to engage in
direct discussions with experimental scientists working in areas that
can
potentially be described by the theories that are under development,
especially in biology. It was felt to have succeeded admirably in these
aims. The sessions on cardiovascular dynamics were especially valuable,
with
data obtained in Lancaster, Ljubljana, Oslo and Pisa being exposed to
general scrutiny and open discussion with theorists expert in e.g.
stochastic dynamics, synchronisation, information theory, relaxation
oscillations, excitable systems and physiology.
The local Director, Professor Aneta Stefanovska did a superlative job in
organising all aspects of the meeting { both scientific and social { the
latter including a tour of central Ljubljana, two excellent concerts, a
magnificent conference dinner and, after the conference, an excursion
taking
in Cerknica (the "disappearing lake"), the Postojna caves, a
wine-tasting,
and another memorable dinner. The meeting was supported by INTAS and by
the
European Science Foundation under its STOCHDYN programme.
Details can be found on the conference web page:
http://osc.fe.uni-lj.si/ljubljana/index.htm
------------------------------------------------------------------------
Report: Modelling of Neuronal Dendritic Trees
ICMS, 14 India St, Edinburgh, 18 June 2004
By Dr Yulia Timofeeva
A one-day meeting on Modelling of Neuronal Dendritic Trees was held on
Friday 18th June at 14 India St. Edinburgh, the birth place of James
Clerk
Maxwell and home to the International Centre for Mathematical Sciences.
Organised by Dr Stephen Coombes (Nottingham) and Dr Gabriel Lord
(Heriot-Watt), the workshop focused on the application of mathematical
tools
from modelling and numerical analysis to information processing and
learning
within a single neuron with branched dendritic structure. The meeting
brought together around 40 researchers in computational neuroscience,
applied mathematics and the life sciences.
Professor John Rinzel (Center! for Neural Science, NYU), started the
meeting
by giving a thorough background about the compartmental method and the
seminal contributions of Wilfrid Rall. He emphasised a minimalistic
approach
to investigating dendritic cable properties using only a few
spatial/electrical compartments. Using a comprehensive compartmental
simulation study, Dr Arnd Roth (UCL) then explained the link between
dendritic morphology and the experimentally observed patterns of forward
and
backward propagating action potentials seen in real neurons. The
usefulness
of mathematical analysis was highlighted by the talk of Professor Steve
Cox
(Rice University, Texas), who described a rigorous approach for
estimating
the location and time course of synaptic input from multi-site potential
recordings. Dr Mickey London (UCL) ill ustrated the success of
information
theory in uncovering the role of dendritic structure in neuronal
computation
and Dr Bruce Graham (Stirling University) spoke about dynamics of
synaptic
integration in a hippocampal pyramidal neuron model. Some new
mathematical
ideas in compartmental modelling were introduced by Dr Ken Lindsay
(Glasgow)
and the meeting concluded with a talk by Dr David Barber (IDIAP,
Switzerland) on a mathematical framework for learning with dendritic
spines.
Overall the workshop was very successful, with a large attendance,
excellent
presentations and fruitful discussions. The meeting was financially
supported by the ICIAM99 fund.
=============================================================================
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