UK Nonlinear News,
Numerical Methods for Bifurcation Problems and
Large-Scale Dynamical Systems
By Eusebius Doedel and Laurette S. Tuckerman (Editors)
The IMA Volumes in Mathematics and its Applications
Volume 119, Springer-Verlag
ISBN: 0-387-98970-6 (Hardcover).
This book is based on the combined proceedings of two
workshops ("Numerical Methods for Bifurcation Problems" and
"Large Scale Dynamical Systems") held as part of the 1997-1998
IMA Academic Year on
Emerging Applications of Dynamical Systems. The book consists of 20
papers covering quite a wide area of research.
As stated at the back of the book
"Several of the papers in this volume treat computational
methods for low- and high-dimensional systems and, in some cases
their incorporation into software packages. A few papers treat
fundamental theoretical problems, including smooth factorization of
matrices, self-organised criticality, and unfolding of singular
heteroclinic cycles. Other papers treat applications of dynamical
systems computations in various scientific fields, such as biology,
chemical engineering, fluid mechanics, and mechanical
A brief summary of the contents of each
paper is given as follows:
- I found the first paper
"Numerical bifurcation techniques for chemical reactor
problems" (by V. Balakotaiah and J. Khinast) particularly
interesting since it strongly relates to my own area of research.
The main thrust of this paper is to
computation of various co-dimension one steady-state and dynamic
singularities for diffusion-reaction, convection-reaction and
diffusion-convection-reaction problems using the shooting technique
and sensitivity functions." The examples provided in this article
were excellent illustrations of the techniques used.
- The second paper is titled "Path-following of large bifurcation
problems with iterative methods" and was written by K. Bohmer,
Z. Mei, A. Schwarzer and R. Sebastian. As the title suggests, it
considers detection and numerical continuation of large dynamical
systems particularly those obtained from discretization of PDEs. The
authors mainly focussed on the reaction-diffusion equation.
- The paper by T.J. Burns, M.A. Davies and C.J. Evans titled
"On the bifurcation from continuous to segmented chip
formation in metal cutting" makes up the third article in this
book. This paper describes how a nonlinear dynamics approach to
modelling the plasticity problem of chip formation during metal cutting
is able to provide new insights into this industrial process. In
particular, the new models suggest that the transition from continuous to
segmented chip formation results from a singular Hopf bifurcation in
the flow of the workpiece material as it is deformed by the cutting
tool. Anyone (like myself) who has a keen interest in industrial
mathematics would find this article to be of particular interest.
- "Using dynamical systems tools in MATLAB" by W.G. Choe and
J. Guckenheimer is the next contribution. The
authors discuss guidelines for building an
interface between Matlab (which possesses
its own rich set of graphics, user interface and numerical
operations) and dynamics tools. The authors describe their
"...efforts as the first steps towards the construction of a
`Dynamics Toolbox' for Matlab...". I am excited at
the possibility of having such a toolbox in Matlab. However I feel
that this article
would be particularly heavy-going for those who are not familiar
- K. Coughlin's article on "Formation and instabilities of
coherent structures in channel flows"
which has a strong fluid-dynamics flavour
makes up the fifth paper
in this book. It discusses
the dynamics of intermittent turbulence in channel flows. The author
describes how the addition of a small forcing term to the Navier-Stokes
equations makes the streamwise vortex solutions become true
equilibria. The latter are then studied by direct numerical
investigation of the relevant linear equations.
- The article "Applications of smooth orthogonal factorizations of
matrices" by L. Dieci and T. Eirola describes
computation of smooth orthogonal factorization of matrices.." are
encountered in dynamical system problems.
The authors also discuss some of the issues that arise during
the implementation of their techniques.
- I found the next article "Continuation of codimension-2
equilibrium bifurcation in CONTENT" (by W. Govaerts,
Yu. A. Kuznetsov and B. Sijnave) to be particularly interesting, since
I have never used the software package CONTENT and this article gave
me the opportunity to learn of its power. (I am an AUTO
user.) This article discusses recent extensions to
CONTENT which enable the user
to compute and continue all co-dimension-2 bifurcations. A
brief discussion on the detection of codimension-3 equilibrium bifurcations
is also given.
- A. J. Homburg's article "Inclination-flips in the
unfolding of a singular heteroclinic cycle" describes the study
of the bifurcations arising from a particular heteroclinic cycle in a two
parameter family of three dimensional vector fields.
- In "
Investigating torus bifurcations in the forced Van Der Pol
oscillator", B. Krauskopf and H. M. Osinga discuss a new
algorithm which they have developed and
implemented in the software package DsTool for computing one-dimensional
stable and unstable manifolds of (Poincare) maps. The authors use
the Van Der Pol oscillator as a test example.
- The tenth article in the book is titled "Quasiperiodic
response to parametric excitations" (by J.M. Lopz and
F. Marques). This article describes a technique
based on Floquet theory and discrete dynamical systems that
can identify the conditions under which a periodically forced system
(susceptible to centrifugal instabilities)
resonates (resulting in resonance horns). It also can identify higher
codimension degenerate bifurcations from space-time resonances.
- I must confess that I knew very little about self-organized criticality
and found the article by J. Lorenz, S. Jackett and W. Qin
bifurcation analysis of periodic solutions of large-scale
very difficult to read initially. If you are like me, I suggest some
background reading of P. Bak's book "How
nature works", Springer-Verlag, New York (1996) before
attempting to read this article.
- The efficient computation (based on Newton-Picard method)
and bifurcation analysis of periodic
solutions of large-scale systems (those arising from discretizing
PDEs) is the basis of K. Lust and D. Roose's article
("Computation and bifurcation analysis of periodic solutions of
- The article "Multiple equilibria and stability of the
North-Atlantic wind-driven ocean circulation"
(by M.J. Molemaker and H. A. Dijkstra) describes a
numerical method (whose code is dubbed
BAGELS by the authors)
which is successful in investigating
multiple equilibria in an idealized finite element ocean model of
the wind-driven ocean circulation in the North-Atlantic -- a large
scale dynamical system.
- The paper by M. Olle ("Numerical exploration of
bifurcation phenomena associated with complex instability")
discusses numerical methods for path-following and detecting
bifurcations in Hamiltonian systems with three or more degrees of freedom.
- Lattice dynamical systems of unbounded medium (sometimes
referred to as coupled map lattices) are the basis of the paper by
D.R. Orendovici and Ya.B. Pesin ("Chaos in travelling waves of
lattice systems of unbound media"). The authors
describe such systems corresponding to some well-known PDEs such as
the reaction-diffusion, Swift-Hohenberg,
Kuramoto-Sivashinsky and Ginzburg-Landau equations. They also discuss
the mechanism for the appearance of spatial and/or temporal chaos
associated with various special classes of solutions.
- The next article by H.G. Othmer, B. Lilly and J.C. Dallon
applies dynamical systems computations to
biology. Their paper titled "Pattern formation in cellular
slime mold" looks at stream formation (which is a prominent
feature of aggregation in low-density fields) in slime mold,
the relationship between cell-based and
continuum descriptions of aggregation, and the origin of streaming in
these two models.
- B. P. Peckham's article "Global parametrization
and computation of resonance surfaces for periodically forced
oscillators" makes up the 17th article in this book.
The title speaks for itself.
- The paper "Computing invariant tori and circles in
dynamical systems" by V. Reichelt discusses several algorithms
(based on the discretization of the graph transform)
which allow the computation of these tori using continuation
strategy. The author shows that these algorithms are faster, more
robust and most importantly more accurate than some existing
The author illustrates these
algorithms on several test examples.
- R. Seydel's article "A design problem for image
processing" is one of my favourites in this book. His
paper outlines the role of bifurcation in image processing. I
find the whole concept of images being processed by
integrating reaction-diffusion equations fascinating and I
will make an effort to look up
some of the papers in Seydel's bibliography.
- The final article in the book is
" Bifurcation analysis for timesteppers" by
L.S. Tuckerman and D. Barkley. The authors discuss a collection of
methods used to
adapt a pre-existing time-stepping code to enable various
bifurcation tasks to be undertaken.
The review of this book was an exhausting and time-consuming precess,
since it involved a mini-review of 20 research articles!
I learnt a lot from this book and will certainly be chasing
a few references. This book is an important and useful
resource which brings together a collection of papers dealing with
bifurcation analysis and large-scale dynamical systems. I highly
it for people who are active in this field of research.
A listing of books reviewed in UK Nonlinear News
UK Nonlinear News thanks
Springer Verlag for
providing a review copy of this book.
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Page Created: 18th January 2001.
Last Updated: 18th January 2001.