Workshop
Organizers Visitors
Postdocs
1. ANNUAL PROGRAM ORGANIZERS
Martin Golubitsky
of the University of Houston, Department of Mathematics is one
of the organizers for the 199798 program year on "Emerging
Applications of Dynamical Systems." He also coorganized
the IMA workshop on "Pattern Formation in Continuous and
Coupled Systems." He writes:
Pattern formation has been studied intensively for most of this
century by both experimentalists and theoreticians, and there
have been many workshops and conferences devoted to the subject.
In the IMA workshop on Pattern Formation in Continuous and Coupled
Systems held May 1115, 1998 we attempted to focus on new directions
in the patterns literature. In particular, we stressed systems
and phenomena that generate new types of pattern (those that
appear in discrete coupled systems, those that appear in systems
with global coupling, and those that appear in combustion experiments)
and on wellknown patterns where there has been significant
recent development (for example, spiral waves and superlattice
patterns).
The participants at this meeting included, in more or less equal
parts, experimentalists and theoreticians. One goal was to continue
communication between these groups, and we were pleased by the
result. Another goal was to familiarize a larger audience with
some of the newer directions in the field, and again the result
was very satisfying.
With these goals in mind, we decided to produce a nonstandard
workshop proceedings. We did not want to publish a collection
of research articles, which could have appeared elsewhere as
refereed journal articles, nor did we want to publish a list
of abstracts. Instead, we attempted to collect a series of minireview
articles of at most 15 to 20 pages (with extensive bibliographies)
that would discuss why certain topics are interesting and merit
additional research. The response has been quite heartening
and we hope that readers will find these reviews a useful entry
into the literature.
Joint work with Dan Luss, University of Houston (Chemical Engineering)
and Steven H. Strogatz, Cornell University (Theoretical and
Applied Mechanics).
John Guckenheimer from Cornell
University, Department of Mathematics was the chair of the organizing
committee for the 199798 year on "Emerging Applications
of Dynamical Systems." His report follows.
I served as chair of the organizing committee for the program
"Emerging Applications of Dynamical Systems" during the 199798
academic year. Thus, spending the year visiting the IMA was
a unique opportunity for me. I think that I also played an important
role in providing mentorship for the postdoctoral fellows in
the program. During the course of the year, I engaged in technical
discussions about problems of common interest with about half
of these fellows, will mentor Kurt Lust at Cornell during 199899
and am continuing a collaboration with Kathleen Rogers and Warren
Weckesser. The remainder of this report will discuss the research
I accomplished and make suggestions for IMA reflecting my role
as chair of the organizing committee.
Research
The main topic for my research during this past year has been
the formulation and implementation of algorithms for computing
periodic orbits of dynamical systems. Periodic orbits, together
with their stable and unstable manifolds are fundamental objects
in the phase space of a dynamical system. Stable periodic orbits
are frequently observed as the limiting behavior of trajectories
computed by numerical integration. For some purposes there are
more effective ways of computing these orbits. Unstable periodic
orbits that cannot be readily observed as the limits of numerical
trajectories are also important in applications. I have been
investigating algorithms that perform direct calculation of
periodic orbits. Such algorithms are constructed in the framework
of boundary value problems for ordinary differential equations.
The primary innovation in our work has been to use a technique
known as automatic or computational differentiation to achieve
very high order accuracy in the methods. The results are impressive.
The new methods achieve higher accuracy with coarser meshes.
They are flexible in their use and straightforward as implementations
of the mathematical problems they solve. This work is complementary
to the work of Kurt Lust, and we have begun discussions about
how to extend our methods to work with large systems.
There are two collaborative projects that I began during the
IMA year. The first is a study of models of two coupled Josephson
junctions, or equivalently two pendula coupled with a torsional
spring. This two degree of freedom conservative mechanical system
has very complex and interesting dynamics. Don Aronson, Sebius
Doedel, Bjorn Sandstede and myself have undertaken extensive
numerical investigations of this system. We have developed a
good understanding of some aspects of how the phase space of
the system is organized. We are working on a paper that will
describe our conclusions.
Kathleen Rogers, Warren Weckesser and I have been studying a
different four dimensional vector field that represents two
coupled oscillators. The system we are studying is a representation
of two neurons coupled through reciprocal inhibition. Each of
the oscillators is a relaxation oscillator with two time scales.
Trajectories within the system evolve on the slower time scale,
with brief periods of rapid transitions that occur on the faster
time scale. As the model parameters of the system are varied,
the patterns of transitions undergo bifurcation. Such phenomena
have been studied as singular perturbation problems, but bifurcation
theory for multiple time scale systems is not yet a highly developed
subject. Thus, our numerical investigations are revealing new
behavior that is interesting both for mathematical theory and
for its biological interpretations.
Organization:
The program was implemented almost exactly as outlined in the
original plans. I think that the set of activities was coherent
and provided an excellent mix of interdisciplinary applications
with emphasis upon the development of mathematical theory and
algorithms. There are only a few comments that I offer for future
improvements in the IMA programs.
The postdoctoral fellows were the focus of most of the IMA program.
The selection committee placed an emphasis upon selecting new
recipients of the PhD. In some cases, the participants did not
complete their theses and other degree requirements until well
into the year. This was a distraction from their ability to
plunge into new projects. It is difficult to predict how long
students will take to complete their degree requirements, but
I recommend that additional attention be given to how IMA can
best deal with this issue. There were six senior visitors at
IMA for the entire academic year. This provided mentorship whose
quality would have been difficult to achieve otherwise. Increased
responsibility from the Minnesota faculty in the programs would
be helpful, especially with the implementation of two year postdoctoral
appointments.
Director's Comment: Starting in fall 1998, a two year postdoctoral
appointment has become the standard. All postdocs are assigned
faculty mentors. The first year of the postdoc experience involves
participation in the IMA theme program. In the second year (when
the IMA theme program may be inappropriate for the second year
postdocs) there is a special seminar for these second year postdocs
(and possibly some small special workshops) with faculty involvement
to continue the mentoring experience. Also there will be a special
teaching development program for some of the postdocs. This
will include participation in the University of Minnesota Bush
Foundation teaching development project that involves oneonone
mentoring of junior faculty by master teachers. For other postdocs
who are interested, there will be opportunities for industrial
interaction as well as teaching mentoring.
There were ten workshops that were included in our original
plan for the year. Despite the long delay between original conception
and the final workshops, the programs were lively and stimulating.
Additional events were added to the program at a later date
and contributed further to making the year's activities. For
each workshop, we set a goal of bringing together groups of
researchers who have not interacted strongly in the past. We
achieved this goal in almost all cases, in some cases superbly.
The final workshop on animal locomotion was noteworthy in having
participants from four communities (dynamical systems, robotics,
biomechanics and electrophysiology) engaged in intense discussions
seeking to build a common understanding of legged locomotion
and swimming. The workshops and tutorials that preceded some
of the workshops were a focal point for the entire year. Despite
IMA guidelines to limit the number of lectures at workshops,
the programs inevitably grew to the point that were was limited
time for informal discussions among the workshop participants
and no time for anything else. Thus, the atmosphere of the IMA
fluctuated from week to week. During some periods, especially
at the beginning and end of the academic year, it was hardly
possible to both attend IMA events and focus upon individual
research. Overall, I think that the amount of time spent in
workshops was good. I do not think it should be increased.
The IMA support staff was very helpful. Nonetheless, there were
some glitches in the communication between the staff and workshop
organizers. These improved during the year, but I recommend
that there be clear policies of what reports will be provided
to the workshop organizers about responses to invitations. I
recommend that a database accessible to organizers of IMA participants
be established to facilitate decisions about meeting programs
after initial invitations are sent. While the organizing committee
was given guidelines for how many invitations should be issued
to different categories of visitors, it was not given information
about budgets or expected numbers of acceptances. This made
it harder to respond to inquiries from potential participants
and difficult to know when the committee should consider recommending
additional invitations. I like the fact that the committees
were not given fixed budgets, but more information about finances
would helped the planning process.
Director's comment: Lists of confirmed workshop participants
are now on the web, publicly available, and are updated regularly.
I think that the IMA could improve the interaction of its programs
with the University of Minnesota, especially outside the mathematics
department. Several of the workshops would have benefited from
having a local member of their organizing committees with responsibility
for encouraging participation of other members of the university
community with the IMA. Conversely, little effort was made by
the IMA to advertise University of Minnesota activities outside
the mathematics department to its members. In particular, the
winter biological sciences segment of the year could have been
enriched substantially by greater involvement with other parts
of the University.
Director's comment: In fact the IMA made considerable efforts
to involve the biological community at the University, including
personal phonecalls, regular mailings of the Newsletter to the
appropriate biological science departments and a large email
list aimed at faculty in these departments, with details of
forthcoming workshops. Despite this, there was little success
in promoting interaction with the biological and medical community
at the University during 199798, a continuation of a longstanding
problem in connecting the science and engineering disciplines
with the biological community. However, during the 199899 year
on Mathematicas and Biology, striking advances were made. More
than 50 reseachers in medicine and biology at the University
took part in the 199899 program. An ongoing MathematicsPhysiology
Seminar was established, with speakers half from math and engineering
department s and half from physiology and medicine. The seminar
is now being upgraded to the McKnight Seminar in Mathematical
Bioscience, hosted by the IMA, cosponsored by the departments
of Neuroscience, Chemical Engineering, Mathematics, and the
Biological Process Technology Institute, and funded by the McKnight
fund of the Graduate School. Hans Othmer, a distinguished mathematician
specializing in Developmental Biology, has just joined the School
of Mathematics and the Digital Technology Center. He will play
an important role in helping the IMA nurture mathbiology links.
Additional joint mathbiology positions, programs and much more
interdiscipinary research effort are clearly in the offing.
2. WORKSHOP ORGANIZERS
Eusebius Doedel of the California
Institute of Technology is one of the organizers of the September
15  19, 1997 workshop on ``Numerical Methods for Bifurcation
Problems." The proceedings for this workshop is combined with
that of a related workshop ``LargeScale Dynamical Systems,
September 29  October 3, 1997" and will appear in the IMA Volumes
in Mathematics and its Applications as Volume 119: Numerical
Methods for Bifurcation Problems and LargeScale Dynamical Systems.
Eusebius Doedel along with Laurette Tuckerman are the editors.
The following preface is written for the combined proceedings:
The papers in this volume are based on lectures given at the
first two workshops held as part of the 19971998 IMA Academic
Year on Emerging Applications of Dynamical Systems. This IMA
Year was organized by John Guckenheimer (chair), Eusebius Doedel,
Martin Golubitsky, Yannis Kevrekidis, Rafael de La Llave, and
John Rinzel. The scientific program had a strong computational
component, as especially reflected in the first two workshops,
which were entirely devoted to computational issues.
Workshop 1, "Numerical Methods for Bifurcation Problems,"
was held in the week of September 1519, 1997. The organizing
committee of this workshop consisted of Eusebius Doedel (chair),
WolfJuergen Beyn, Bernold Fiedler, Yannis Kevrekidis, and Jens
Lorenz. The workshop concentrated on complex computational issues
in dynamical systems. While computational techniques for lowcodimension
local bifurcations in fewdegree of freedom systems are in advanced
state of development, much work remains to be done on the numerical
treatment of higher codimension singularities. More importantly,
there is a pressing need for the development of numerical methods
for computing global objects in phase space, their interactions
and bifurcations. This workshop brought together mathematicians,
numerical analysts, and computer scientists working on these
problems. Particular topics included the detection of bifurcations
and the development of associated numerical and visualization
software. Also considered were important theoretical issues,
such as smooth factorization of matrices, selforganized criticality,
and singular heteroclinic cycles. The numerical computation
of manifolds, such as invariant tori and resonance surfaces
were also studied.
Workshop 2, "Large Scale Dynamical Systems," was held
during the week of September 29October 3, 1997. It was organized
by Laurette Tuckerman (chair), Edriss Titi, Herbert Keller,
and Don Aronson. The numerical study of lowdimensional dynamics
in large scale sets of ODEs and discretizations of PDEs necessitates
the development of special purpose algorithms for simulations,
stability and bifurcation analysis. This workshop addressed
the development and application of special iterative methods
for large scale systems. It also considered global model reduction
schemes for PDEs. A related goal is to encourage the interpretation
of largescale physical problems as dynamical systems which,
although highdimensional, undergo lowcodimension bifurcations.
Applications of special interest include selected problems arising
in fluid flows, and pattern formation in reactiondiffusion
systems.
We would like to thank the IMA and the program coordinators
for holding this workshop. We thank outgoing and incoming directors
Avner Friedman and Willard Miller, and especially Robert Gulliver
for coordinating the workshops, and the IMA staff for providing
logistic support. We also thank Patricia V. Brick for her important
contribution to this volume as editorial and production coordinator
at the IMA.
Rafael de la Llave of the
Department of Mathematics, University of TexasAustin is the
main organizer of the workshop on "Dynamics of Algorithms"
held on November 1721, 1997. The proceedings for this workshop
appears in the IMA Volumes in Mathematics and its Applications
as Volume 118. Linda R. Petzold of the Department of Mechanical
and Environmental Engineering, University of CaliforniaSanta
Barbara and Jens Lorenz of the Department Mathematics and Statistics,
University of New Mexico serve as coeditors Below is the preface
for the book:
Algorithms and dynamics reinforce each other since iterative
algorithms can be considered as a dynamical system: a set of
numbers produces another set of numbers according to a set of
rules and this gets repeated. Issues such as convergence, domains
of stability etc. can be approached with the methods of dynamics.
On the other hand, the study of dynamics can profit from the
availability of good algorithms to compute dynamical objects.
Fundamental concepts such as entropy in dynamical systems and
computational complexity seem remarkably related. This interaction
has been apparent in the study of algorithms for numerical integration
of ordinary differential equations and differential algebraic
equations from the beginning (Newton already worried how to
compute numerical solutions of ODE's) and in other areas such
as linear algebra, but it is spreading to more areas now, and
deeper tools from one field are being brought to bear on the
problems of the other.
This collection of papers represents the talks given by the
participants in a workshop on "Dynamics of Algorithms"
held at the IMA in November 1997. We hope that it can give a
feel for the excitement generated during the workshop and that
it can help to further the interest in this important and growing
area full of fruitful challenges.
Laurette Tuckerman came
with her husband, Dwight Barkley for the entire program year.
She is one of the organizers of the IMA Workshop on LargeScale
Dynamical Systems held on September 29  October 3, 1997. Her
report follows:
I.
Open sheardriven flows
A.
Perturbed plane Couette flow
I collaborated with Dwight Barkley (yearlong IMA visitor for
19978) on simulations of perturbed plane Couette flow. Plane
Couette flow, the flow between two parallel plates translating
in opposite directions, has long been known to be linearly stable
at all Reynolds numbers, but to undergo a sudden transition
to threedimensional (3D) turbulence in the laboratory and in
numerical simulations. In a search for intermediate states which
might explain the transition mechanism, the experimental group
of Bottin et al. inserted a thin wire into the flow, and observed
3D steady states. Barkley and I were able to numerically simulate
these states and to determine that they arose from a subcritical
pitchfork bifurcation. We have since been investigating the
dependence of the scenario on wire radius, as well as the transition
from these steady states to timedependence and turbulence.
B.
Symmetries in cylindrical wake flow
Dwight Barkley (yearlong visitor) and Ron Henderson (participant
in IMA workshop on LargeScale Dynamical Systems in SeptemberOctober
1997) numerically calculated the 3D instabilities of the periodic
2D flow in the wake of a cylinder. Two instabilities were found,
called mode A and mode B, with quite different wavelengths and
symmetry properties. Experimental evidence suggests that these
two modes may interact, with resulting complex dynamics. Guided
by Martin Golubitsky (IMA visitor in May 1998, coorganizer
of workshop on Symmetry and of the year on Emerging Applications
of Dynamical Systems), I began to undertake an analysis of the
interaction of these two modes in terms of bifurcation theory
in the presence of symmetry. This required mastering the concepts
of invariant and equivariant normal forms, isotropy lattices
and isotypic decompositions. Of great help in doing so was an
informal study group organized by Warren Weckesser (IMA postdoc,
19978) in the Spring of 1998 on this very subject. Because
the 2D cylindrical wake flow is periodic, the 3D stability analysis
carried out by Barkley and Henderson is a Floquet analysis,
and the symmetries are spatiotemporal, rather than purely spatial.
In fact, one of the main focuses of the IMA workshop on Symmetry
in May 1998 became the framework for analyzing spatiotemporal
symmetry very recently developed by participants and speakers
Peter Ashwin, Jeroen Lamb, Ian Melbourne, and Alistair Rucklidge.
II.
Closed convective flows
A.
Numerical work
I continued work on convection in various configurations. With
Patrick Le Quéré and Shihe Xin, I completed an
article on convection driven by equal and opposite horizontal
thermal and concentration gradients. This turned out to involve
a subcritical circle pitchfork followed by a supercritical drift
pitchfork in the case of a vertically periodic cavity and a
transcritical bifurcation in a square cavity. With Daniel Henry
(IMA visitor during SeptemberOctober 1997) and Alain Bergeon
(participant in IMA workshop on LargeScale Dynamical Systems),
I completed a survey article on surfacetensiondriven (Marangoni)
convection due to combined thermal and concentration gradients.
Henry, Bergeon, and I also continued work on Marangoni convection
in threedimensional rectangular containers, and we discussed
the interpretation of this work in the context of symmetries
of rectangles and squares with Edgar Knobloch (IMA visitor during
May 1998) and Gabriela Gomes (yearlong IMA visitor during 19978).
B.
Analytic work
Convection driven by competing thermal and concentration gradients
was extensively studied in the 1980's as a physically realizable
prototype of a codimensiontwo point, at which a curve of Hopf
bifurcations was annihilated by joining a curve of steady bifurcations,
accompanied by the existence of heteroclinic infiniteperiod
cycles. These curves describe the critical Rayleigh number (temperature
difference) for convection as a function of separation parameter
(ratio of solutal to thermal effects).
I discovered that these bifurcation curves, i.e. the linear
stability diagram, could all be derived from the following property:
The eigenvalues of a 2 × 2 matrix whose entries depend
linearly on a control parameter undergo either avoided crossing
or complex coalescence, depending on the sign of the coupling
(product of the offdiagonal terms) near the point at which
the diagonal terms intersect. My interpretation would organize
binary convection around the case of zero separation parameter,
at which the coupling vanishes and the eigenvalues simply cross
transversely.
More surprising was the realization that the nonlinear properties
could also be explained in this way. The structure of the equations
turns out to be such that the system of coupled nonlinear equations
describing the steady states reduce to a 2 × 2 eigenvalue
problem with the square of the convection amplitude as eigenvalue.
The structure of the matrix is almost identical to that governing
the linear stability; only the interpretation changes. For instance,
complex coalescence for the nonlinear problem must be interpreted
as a saddlenode bifurcation (disappearance of a pair of solutions)
instead of as the onset of oscillatory behavior. In developing
this framework, I benefitted greatly from conversations with
John Guckenheimer (program organizer and yearlong visitor),
Edgar Knobloch (visitor, May 1998), and Fritz Busse and Hermann
Riecke (speakers at the May 1988 workshop on Symmetries).
Publications
Written and submitted while at IMA
D. Barkley and L.S. Tuckerman, Stability analysis of perturbed
plane Couette flow, submitted to Phys. Fluids.
D. Barkley and L.S. Tuckerman, Linear and nonlinear stability
analysis of perturbed plane Couette flow, in Proceedings
of the Seventh European Turbulence Conference, ed. by U. Frisch
(Kluwer Academic Publishers, Dordrecht, 1998).
L.S. Tuckerman and D. Barkley, Bifurcation analysis for timesteppers,
in Numerical Methods for Bifurcation Problems and LargeScale
Dynamical Systems ed. by E. Doedel, B. Fiedler, Y. Kevrekides,
W.J. Beyn, J. Lorenz, L.S. Tuckerman, E. Titi, H.B. Keller,
and D. Aronson (Springer, New York, to appear).
Revised
while at IMA
S. Xin, P. Le Quéré, and L.S. Tuckerman, Bifurcation
analysis of doublediffusive convection with opposing horizontal
thermal and solutal gradients Phys. Fluids. 10, 850858
(1998).
A. Bergeon, D. Henry, H. BenHadid, and L.S. Tuckerman, Marangoni
convection in binary mixtures with Soret effect, J. Fluid
Mech., in press.
Researched
at IMA, now being written
L.S. Tuckerman, D. Henry, and A. Bergeon, Binary fluid convection
as a 2 by 2 matrix problem, to be submitted to Physica D.
A. Bergeon, D. Henry, H. BenHadid, and L.S. Tuckerman, Threedimensional
Marangoni instability pattern selection, to be submitted
to J. Fluid Mech.
Annual Program Organizers
Workshop Organizers
Postdocs
3. VISITORS/SPEAKERS
Dwight Barkley of University
of Warwick, Mathematics Institute was one of the longterm visitors.
He submits the following report:
My work at the IMA falls into two broad classes which I shall
describe separately.
I.
Waves in excitable media.
I completed a new fast computer program for simulating and visualizing
in real time waves in threedimensional excitable media. Two
examples of such media are the BelousovZhabotinsky chemical
reaction and cardiac tissue. It has been known for some time
that weak parametric forcing provides a simple method for control
of spiral waves in twodimensional excitable media. Essentially
nothing was known, however, about the effect of parametric forcing
in threedimensional excitable media where solutions are much
richer. Dr. Rolf Mantel (IMA postdoc) and I investigated in
the effect of weak parametric forcing of two of the most fundamental
structures in three dimensions: the socalled axisymmetric scroll
ring and the twisted scroll ring. We determined to high accuracy
the spatiotemporal dynamics of both structures with and without
forcing. Most significantly we were able to understand the dynamics
through normal form equations based on noncompact symmetry
groups. This understanding benefited tremendously from the conversations
with several visitors to the IMA (B. Sandstede, I. Melbourne,
B. Fiedler, and C. Wulff). The work has been submitted for publication
[1].
II. Instabilities and dynamics of fluid flows.
Collaborations
took place with both long and short term IMA visitors on several
problems in fluid dynamics. L. Tuckerman and I performed extensive
numerical linear and nonlinear stability computations of a classical
shear flow, socalled plane Couette flow, perturbed with small
ribbon in the center of the flow. We were able to confirm that
experimentally observed streamwise vortices in this fluid flow
resulted from a subcritical symmetry breaking bifurcation of the
basic laminar flow [2]. We found transition to temporally complex
states, signifying the onset of a weak form of turbulence. We
are currently pursuing this line of research to understand the
scenario through which the flow becomes turbulent as the size
of geometric perturbation in decreased to zero.
I collaborated with L. Tuckerman in her work on developing a
suite of largescale bifurcation methods based in timestepping
algorithms [3].
IMA visitors M.G.M. Gomes, R. Henderson, and I studied threedimensional
transition in the separated flow generated by a sudden expansion
in an otherwise parallel flow channel. This geometry is known
as a backwardfacing step. We found that the first instability
encountered as the Reynolds number is increased leads directly
to a threedimensional state, and that surprisingly the flow
remains linearly stable to twodimensional disturbances for
very large Reynolds number [4].
I developed a heuristic description of nonlinear threedimensional
flow patterns in the wake of flow past a circular cylinder [5].
A collaboration was begun with IMA visitors L. Tuckerman and
M. Golubitsky in which we plan to extend this work with a combination
of symmetric bifurcation theory for bifurcations from periodic
orbits and largescale computer simulations of the governing
fluid equations. Based on the initial work done while at the
IMA we have successfully obtained a grant of over 1600 hours
of supercomputer time at IDRIS (France) for performing the necessary
computations. We expect that a number of other fluid flows might
also be examined in this way in future years.
References:
[1] R.M. Mantel and D. Barkley, "Parametric forcing of
scrollwave patterns in threedimensional excitable media,"
Physica D (submitted).
[2] D. Barkley and L.S. Tuckerman, "Stability analysis
of perturbed plane Couette flow," Phys. Fluids (submitted).
IMA preprint 1545.
[3] L.S. Tuckerman and D. Barkley, "Bifurcation analysis
for Timesteppers," IMA preprint 1564.
[4] D. Barkley, M.G.M. Gomes, and R.D. Henderson, "Threedimensional
instability in flow over a backwardfacing step" in preparation
for J. Fluid Mech.
[5] D. Barkley, "Nonlinear stability theory for threedimensional
wake transition," in proceedings of the ASME International
Fluids Engineering Division Summer Meeting, Washington, D.C.
June 1998.
Fernanda Botelho from the
University of Memphis, Department of Mathematics was in the
IMA from January 10  May 31, 1998. She expresses the following:
"
I want to thank you for this opportunity. My stay has been most
challenging intellectually while profiting from a great work
environment. I feel, I return to the University of Memphis in
a new and better professional mode, thanks to this productive
sabbatical leave at your Institute. Finally, let me add a word
about your staff. I was always very impressed by the efficient
way all my logistics issues were dealt with, this ranging from
housing information to T_{E}X and computer questions
as well as local entertainment. Everyone was very helpful. Definitively,
IMA is one of the best places I have ever been."
Gene Cao
of Michigan Technological University, Department of Mathematical
Sciences attended the workshop: Algorithmic Methods for Semi
Conductor Circuitry, November 2425, 1997 and the special Workshop:
Knowledge and Distributed Intelligence (KDI) Opportunities in
the Mathematical Sciences, March 7, 1998. He has the following
impression:
It is certainly a successful workshop, especially in bringing
researchers from industry/EE Depts to interact with each other.
They seem so happy with the workshop that they will organize
another one next year even without IMA's involvement/support.
It would be even better for them as well as for mathematicians,
however, that more attention is paid to get mathematicians involved.
It was a big commitment for me to attend this workshop (under
the $1500/year travel budget. Industrial participants may not
have such constraint for math faculty, according to a Bell Labs
member). I have to confess that I was a little bit disappointed
since it seem too similar to an engineering workshop.
Benoit Dionne,
University of Ottawa, Department of Mathematics and Statistics
was one of the long term participants. The following summarizes
his research activities during his visit at from September 1
to October 31, 1997.
During my visit at the IMA, I worked on a project with Martin
Krupa (Technical University of Vienna) who was also visiting
the IMA in the Fall 1997. We started on this project shortly
before arriving at the IMA. The project is still not complete
but we hope to complete it soon.
In this project, we study period doubling in arrays of identically
coupled identical cells. Each cell is a system of coupled Josephson
junctions. The typical symmetry group of the system of differential
equations governing the array of cells is the wreath
product of a subgroup of permutations on the cells (global symmetry)
and the permutations on the Josephson junctions in each cells
(each cell has the same internal symmetry). The Equivariant
Branching Lemma for period doubling of mappings is used
to determine the existence and symmetry of each branches of
solutions emanating at a period doubling bifurcation point.
The Equivariant Branching Lemma for period doubling of mappings
is applied to the Poincare map associated to a periodic
solution that has the full symmetry.
I also had the opportunity during my visit at the IMA to meet
Sebius Doedel (Concordia University). A possible collaboration
may come out of our discussions. The project that I have in
mind will be to add to auto (a software developed by
Sebius Doedel to follow branches of solutions of differential
equations) the functionality to do branching at period doubling
bifurcation points in systems with symmetries. I should be on
leave next Fall and I hope to visit Sebius Doedel at that time.
Bernold
Fiedler
of Free University of Berlin, Instiute of Mathematics was a
long term visitor. He comments on some research activities he
undertook during his fall 1997 stay at the IMA:
Sandstede and Scheel have proved a result concerning Hopf bifurcation
from constant speed travelling waves to travelling waves with
oscillating wave speeds. Their result is the first to account
for Hopf bifurcation from the continuous spectrum.
In the seminar "Josephson Junctions," organized by
Aronson and Doedel, significant progress was made concerning
topologically nontrivial heteroclinic orbits.
D. Lewis and B. Fiedler discussed the behavior of discretization
schemes near relative equilibria to compact and noncompact
group actions. It turns out that certain discretization schemes
are particularly well adapted to the computation of secondary
symmetry breaking bifurcations from relative equilibria.
The geometry of intersections of vortex filaments of threedimensional,
timedependent scroll wave patterns in excitable media was investigated,
both analytically and numerically, by the IMA PostDoc R. Mantel
and B. Fiedler. Paper & video are in preparation.
During a visit of B. Fiedler to participating institution UW,
Madison, progress was made with S. Angenent concerning stationary
versus heteroclinic blowup of maximal compact invariant sets
in scalar reaction diffusion equations.
With J. Alexander at participating institution UMD, College
Park, B. Fiedler has clarified the two simplest possibilities
of transversely nondegenerate Hopftype bifurcation from a degenerate
line of equilibria. Such situations arise in certain graphs
of linearly coupled oscillators. R. Pego has pointed out a relation
to (spurious) binary oscillations in certain discretization
schemes for systems of conservation laws in one space dimensions.
Particularly helpful were additional discussions concerning
coordinate blowup and slowfast singular perturbation decompositions
with P.K.R.T. Jones.
With I. Melbourne at participating institution U of Houston,
Texas, B. Fiedler discussed new possibilities for an emerging
normal form theory of vector fields near relative equilibria
to noncompact group actions. An application is bifurcation from
twisted scroll waves to genuinely threedimensional, nonplanar
waves travelling with oscillating wave speeds along a periodically
wobbling axis. The associated circular vortex filament, linked
to the axis of propagation, will undergo periodic shape changes
which preserve linking. With careful experiments just emerging,
these mathematical predictions are ahead of observations, in
this case.
Bernold Fiedler also took part in an informal seminar as he
describes below.
"Continuous
Spectra"
Continuous Spectra arise naturally in linear partial differential
equations on unbounded domains. Traditional areas include hyperbolic
wave equations, Schroedinger operators, scattering theory, etc.
More recently, a variety of nonlinear wave phenomena and patterns
have been investigated, both analytically and numerically, in
the context of semilinear reaction diffusion equations. From
an applied point of view, patterns in the BelousovZhabotinsky
reaction are a primary source of inspiration: travelling waves,
target patterns, spiral waves, meandering spirals have been
observed. Other applications with similar phenomeno logy include
convection patterns in fluids and COoxidation on platinum monocrystals.
The seminar started with a thorough review of functional analysis
results on (various types of) continuous spectra and their per
turbation properties. Results on travelling waves and their
continuous spectra were reviewed next. Progress was made in
the understanding of spectra both in the infinitely extended
discrete case and the continuous limit. Behavior under truncation
was also discussed.
A new result by Bjoern Sandstede and Arnd Scheel was presented,
which addresses Hopf bifurcation from a pulse type travelling
waves due to the continuous spectrum crossing the imaginary
axis.
Finally, the issue of spectral intervals appearing in the bifurcation
of higherdimensional tori was reviewed by George Sell.
I would like to thank all participants for their active interest
and for the pleasant and inspiring atmosphere of this notsoplanned
seminar in a wonderful working environment.
Wulfram Gerstner
of Swiss Federal Institute of Technology Lausanne, Centre for
Neuromimetic Systems was a participant in the workshop: Computational
Neuroscience held on January 1423, 1998. He writes:
This is a short note just to say how much I liked my stay at
the IMA during the workshop on computational neuroscience in
January. I thought it was a great workshop which took place
in an environment which was just perfect for such an event.
Thanks to all of you who put in so much effort to make things
run smoothly.
M. Gabriela M. Gomes
of Universidade do Porto was a longterm participant. Following
is a direct quotation from her.
"
I am looking forward to my next visit to the IMA in May. Let
me add that 1997/98 at the IMA was a very good year for me,
and I would like to thank the IMA again for having invited me."
Below is her report on research related to visit to the IMA
in 1997/98.
 Project
1: Threedimensional instability in flow over a backwardfacing
step (with Dwight Barkley and Ronald Henderson) We performed
a threedimensional computational stability analysis of flow
over a backward facing step. The analysis shows that, as the
Reynolds number is increased, the first absolute linear instability
of the steady twodimensional flow is a steady threedimensional
bifurcation. Stability spectra were obtained for representative
Reynolds numbers. (This project was partially carried out
while Dwight Barkley and myself were visiting the IMA in 1997/98.
The use of the IMA computer facilities was crucial in obtaining
the final stability results and flow visualizations.)
 Project
2: Spatial hidden symmetries in pattern formation (with
Isabel Labouriau and Eliana Pinho) IMA Preprint Series 1582,
August 1998 Partial differential equations that are invariant
under Euclidean transformations are traditionally used as
models in pattern formation. These models are often posed
on finite domains (in particular, multidimensional rectangles).
Defining the correct boundary conditions is often a very subtle
problem. On the other hand, there is pressure to choose boundary
conditions which are attractive to mathematical treatment.
Geometrical shapes and mathematically friendly boundary conditions
usually imply spatial symmetry. The presence of symmetry effects
that are "hidden" in the boundary conditions was
noticed and carefully investigated by several researchers
during the past 1520 years. Here we review developments in
this subject and introduce a unifying formalism to uncover
spatial {\em hidden symmetries} (hidden translations and hidden
rotations) in multidimensional rectangular domains with Neumann
boundary conditions. (This review was written during my visit
to the IMA in 1997/98.)
 Project
3: Blackeye pattern: a representation of three dimensional
symmetries in thin domains The first experimental evidence
for Turing patterns was observed in the CIMA reaction by De
Kepper and colleagues. Ouyang and Swinney performed further
experiments in a "thin" layer of gel. Patterns observed
at onset were basically twodimensional. However, beyond onset
a structure that does not typically occur in twodimensional
domains was observed  the blackeye pattern. In this letter
we use the full threedimensionality of the patterned layer
to find a setting where blackeye patterns naturally occur.
We propose that blackeye patterns have the symmetry of a
body centered cubic lattice. (This research was initiated
during my visit to the IMA in 1997/98. Discussions with other
visitors, including workshop participants, were very helpful.
My recent visit to the IMA in September 1998 was also of relevance
to this project.)
 Project
4: Symmetry and symmetrybreaking approches to strain
formation in pathogen populations The antigenic diversity
exhibited by many pathogens motivated the construction of
mathematical models describing the interaction of a large
number of strains. Depending on the particular pathogens,
two strains can either act by inhibition (crossimmunity)
or by enhancement. The nonlinear differential equations modeling
such systems can achieve a high level of complexity which
hides the underlying features of the system. By introducing
a set of plausible symmetry assumptions, we provide the systems
with a structure that powerful group theoretical tools can
handle. These approaches provide a static view of pathogen
evolution. From an evolutionary perpective, a natural set
of speculative questions which will be addressed is: What
is the mechanism responsible for strain formation? How different
do the pathogens have to be in order to be classified into
different strains? Are new strains created indefinitely? What
happens to the old ones? Is there a limit to the number of
different strains that can cocirculate? We will try to answer
some of these questions by modeling the mechanism of strain
formation as a symmetrybreaking bifurcation. Contact with
field work will be maintained through this work. (This project
was identified during my visit to the IMA in 1997/98, and
will be carried out in the University of Warwick. I will visit
the IMA again in May 1999 to attend tutorials and workshops
in epidemiology.)
Daniel Henry of Ecole Centrale
de Lyon participated in the IMA Tutorial on Numerical Methods
for Bifurcation Problem. He shares the following:
 Introduced
by Laurette Tuckerman to Gabriela Gomes, we had the project
(Alain Bergeon and me) to collaborate with her on the problem
of hidden symmetries. We had done some computations in a 3D
MarangoniBenard situation and it seemed interesting to do
some extra computations with different boundary conditions
and with a certain size of box, in order to analyse the structure
of the solutions. But in fact my stay in Minneapolis was too
short to begin practically on the subject, and since my return
I was too busy with administrative and educational tasks.
 Before
coming to Minneapolis, I got the acceptations from European
scientific associations to organize a workshop on "Continuation
Methods in Fluid Mechanics." One of the coorganizers
was Laurette Tuckerman. During my stay in Minneapolis, we
had the opportunity to meet H. Keller and E. Doedel, and so
we decided to invite them to our workshop as invited speakers.
This workshop will take place in France in Aussois (the Alps)
in September 1998.
Mike Jolly from Indiana
University, Department of Mathematics was a one year visitor.
Below is his report:
1.
Accurate Computation of Inertial Manifolds (with R. Rosa
and R. Temam)
We have implemented an algorithm developed by Rosa [6] to compute
inertial manifolds to arbitrary accuracy. This approach differs
from that of most approximate inertial manifolds in that convergence
can be achieved with the dimension held fixed. The algorithm
was tested on an ODE for which we know an exact inertial manifold.
This example serves to demonstrate how to choose certain algorithm
parameters to optimize the convergence. We also applied the
algorithm to the KuramotoSivashinsky equation, and carried
out an analysis of the effect of truncating the higher modes
for PDE cases such as this. Finally the algorithm was adapted
to compute inertial manifolds with delay and its efficiency
compared to a shooting method. A paper on this work is nearly
complete. We plan to submit this an IMA preprint, as well as
for publication.
2.
Accurate Computation of Center Manifolds (with R. Rosa)
We have adapted the algorithm described above to compute center
manifolds. We have validated the code on some simple ODEs, and
plan to illustrate how it can be advantageous over the traditional
method of local approximation by Taylor expansion by considering
some cases where the manifold is not smooth. We will restrict
the scope of a paper on this work to the ODE case, in order
to reach a wide audience.
3. Computation of Solutions to an Elliptic Boundary Value Problem
on an Infinite Cylinder. (with R. Rosa and E. Titi)
We are applying the algorithm described above, but now in a
PDE context. Kirchgassner [4] showed that for this problem there
exists a twodimensional center manifold. Some analysis is required
to ensure that all conditions are met for convergence. This
is a computationally intensive application. A code has been
developed, and some preliminary analysis carried out. We will
compare to work by Fuming Ma [5], which used a different method
to compute the manifold for this particular problem.
4.
Evaluation of Dimension Estimates for Inertial Manifolds of
the KuramotoSivashinsky Equation
Up to now estimates for this dimension have typically been of
the form dim
cL^{b}, where c is a universal constant
and L is the length of the domain. Over the last decade
there has been a dramatic reduction of the exponent b,
yet c remains an elusive quantity, the end result of
a series of transformations of other universal constants after
considerable analysis. The purpose of this work is to determine
(and to some extent reduce) such universal constants and thereby
arrive at a means to calculate the dimension of an inertial
manifold. This was done by reworking the analysis of Collet
et al. [1] to determine the radius if an absorbing ball,
and then that of Temam and Wang [7] to determine the dimension
of an inertial manifold. The numbers in the end are quite large,
compared to what we speculate from computational evidence. We
then calculated how the dimension would vary if the estimate
for the absorbing ball could be reduced, to see what it might
take to make the rigorous dimension close to what we conjecture.
A paper on this work is nearly ready to turn in as an IMA preprint,
and submit for publication.
5.
Computing Invariant Manifolds by Evolution (with J. Lowengrub)
A convenient method for the computation of global (un)stable
manifolds for ODEs is to evolve the boundary of a local (un)stable
manifold. This approach allows for the capture of global manifolds
which fold over in a complicated fashion. The basic difficulty
in computing in this way manifolds of dimension two and higher
is that different growth rates will cause trajectories representing
the manifold to ultimately be concentrated in the fastest direction.
Guckenheimer and Worfolk [3] get around this by eliminating
the flow in the direction tangent to the curve one is evolving.
The effect is to generate geodesic curves on the manifold, and
in many cases this will result in a good representation of the
manifold. Yet there are certain situations where such geodesic
curves would miss large portions of the manifold. We have developed
a method which replaces the tangential flow with one that preserves
equal arclength representation of the curve of evolution. This
is tantamount to evolving a PDE, which seems to pose some interesting
computational challenges of its own. The computations for this
project are nearly complete, and we will soon begin to write
up the results.
6.
Estimate of an Effective Viscosity Generated by Iterated Approximate
Inertial Manifolds (with C. Foias and Oscar Manley)
Approximate inertial manifolds allow for the approximation of
an evolutionary equation such as the NavierStokes equation
(NSE) to an equation governing the evolution of only the low
modes. In an earlier work, Foias, Manley and Temam [2] showed
that that a certain approximate inertial form (AIF) can
be put into the same form as the original NSE, with nonlinearity
enjoying the same orthogonality condition to ensure dissipativity.
In doing so the viscosity is modified, and becomes dependent
on the velocity. The process however can be repeated on the
AIF, an indefinite number of times, resulting in final AIF which
has the same nonlinearity as the first AIF, but now an infinite
number of terms in the "effective" viscosity. We have since
obtained a bound for the effective viscosity in most of phase
space and interpret a flow across any regions where the effective
viscosity is infinite as being a purely linear flow with infinite
speed. We have also explored a similar iterated procedure using
a better AIF at each stage. Finally we have derived a new recurrent
estimate for the high modes of the exact solution of the NSE.
The last two results were obtained while the three investigators
were together at the IMA in April, 1998. We will put all the
work together for an IMA preprint and eventual submission for
publication.
7.
Visualizing Global Bifurcations of the KuramotoSivashinsky
Equation (with M. Johnson and I. Kevrekidis)
We have used the visualization of twodimensional stable and
unstable manifolds to understand a connection between two seemingly
unrelated global bifurcations. They involve two heteroclinic
connections, one of Silnikov type, the other triggered by the
collision of two manifolds. One amusing aspect of this work
is that the images are strikingly similar to those of a famous
archeological find of a Viking ship (The Oseberg), so much so
in fact that we are using nautical terminology for many complicated
dynamical objects to make the presentation simpler. A paper
on this is nearly ready to submit as an IMA preprint, as well
as for publication.
8. Using Inertial Manifolds in the Computation of Lyapunov Exponents
(with Erik Van Vleck)
We have outlined a plan to compute Lyapunov exponents using
inertial manifolds. We expect that the extra work in computing
the flow on the manifold will be more than offset by the reduction
in the size of the associated linear system which must be evolved
to compute the exponent. The conception of project came as a
result of our interaction at the IMA.
References
[1] Pierre Collet, JeanPierre Eckmann, Henri Epstein, and Joachim
Stubbe. A global attracting set for the KuramotoSivashinsky
equation. Comm. Math. Phys., 152(1):203214, 1993.
[2] Ciprian Foias, Oscar P. Manley, and Roger Temam. Iterated
approximate inertial manifolds for NavierStokes equations in
2D. J. Math. Anal. Appl., 178(2):567583, 1993.
[3] John Guckenheimer and Patrick Worfolk. Dynamical systems:
some computational problems. In Bifurcations and periodic orbits
of vector fields (Montreal, PQ, 1992), volume 408 of NATO Adv.
Sci. Inst. Ser. C Math. Phys. Sci., pages 241277. Kluwer Acad.
Publ., Dordrecht, 1993.
[4] Klaus Kirchgassner. Wavesolutions of reversible systems
and applications. J. Differential Equations, 45(1):113127,
1982.
[5] Fu Ming Ma. Numerical approximation of bounded solutions
for semilinear elliptic equations in an unbounded cylindrical
domain. Numer. Methods Partial Differential Equations, 9(6):631642,
1993.
[6] R. Rosa. Approximate inertial manifolds of exponential order.
Discrete and Continuous Dynamical Systems, 1:421448, 1995.
[7] Roger Temam and Xiao Ming Wang. Estimates on the lowest
dimension of inertial manifolds for the Kuramoto{S}ivashinsky
equation in the general case. Differential Integral Equations,
7(34):10951108, 1994.
Juergen Moser
of Fachinformationszentrum Karlsruhe, Production Division was
a guest of the School of Mathematics and had an office in the
IMA. He reports:
During my visit , April 130,1998 at the University of Minnesota
I had scientific contact with various members at the Department
of Mathematics, the IMA as well as the Geometry Center. I gave
1) a Colloquium lecture "Dynamical systems and the viscosity
solutions of the HamiltonJacobi equation" and 2)a seminar
talk "A Lagrangian proof of the invariant curve theorem
for twist mappings" (R. Moeckel can provide you with the
details). A manuscript (jointly written with H. Jauslin and
H.O. Kreiss) conerning the first lecture was distributed at
the time, and a preliminary preprint (jointly with M.Levi) also
was left at the IMA. Both topics led to interesting discussions
with visitors as well as with permanent members. The first topic
was motivated by the goal for constructive methods for finding
invariant tori for Hamiltonian systems, methods which can be
numerically implemented. This leads to nonlinear partial differential
equations, which are modified to parabolic differential equations
by adding an artificial viscosity term. We considered, in particular,
the model case of the Burgers equation with an added periodic
forcing term and asked for periodic solutions. They can be obtained
as asymptotic limit, as the time goes to infinity, from the
any solution of the initial value problem. One question is to
find quantiative information about the rate of this asymptotic
approach, a problem about which H. Weinberger and I had fruitful
discussions. We could establish that this rate is exponential
in time, but it remains to study the dependence of the exponent
in terms of the viscosity coefficient. This leads to a Harnack
inequality for a linear parabolic differential equation, where
one needs quantitative information about the relevant constant.
Numerical experiments indicate a linear dependence. About nonlinear
parabolic differential equations I learned interesting ideas,
especially about the analyicity of the solutions from (my roommate)
Titi, connected with the methods developed by Foias. Also with
D. Sattinger I had a valuable exchange about the solutions of
the Kortewegde Vries equation, especially those solutions whose
initial values are given by elliptic or hyperelliptic functions,
and his numerical experiments. These discussions did not lead
to final results, and I was the one who profitted from them.
R. Moeckel explained his interesting work on the nbody problem,
trying to find connecting orbits between unstable configurations.
With anumber of younger mathematicians I discussed the new proof
of the twist theorem, presented at my seminar lectures. At the
Geometry Center R. McGehee and Eduardo Tabacman were very helpful
in providing computer graphics relevant for dynamical systems.
I plan to use these in my plenary lecture at the International
congress ICM 98 in Berlin. It goes withought saying that I had
many mathematical discussions with other other guests, students
and faculty members, such Guckenheimer. Foias, A. Friedman,
Aronson, Serrin. The visit was indeed fruitful for me, and hopefully
also for the institute.
Yakov Pesin
of Penn State University, Department of Mathematics was a visitor
from September 27  October 2. He shares the following:
During my visit I worked with M. Jiang (a visitor of the Institute)
and we completed the paper: "Equilibrium measures for coupled
map lattices: existence, uniqueness, and finitedimensional
approximations."
The paper is accepted for publication in Comm. of Math. Phys.
and acknowledgement to the IMA is gladly expressed.
Let me use this oportunity and thank you again for the warm
hospitality that I received at the Institute.
Fernando Reitich
of the University of Minnesota, School of Mathematics reports
the following:
Our IMArelated research activities over the past academic year
were mainly focused on initiating a research program in Mathematical
Biology, in preparation for the upcoming year at IMA. Due in
part to our experience in the mathematics of materials (which,
incidentally, was greatly enhanced by our participation in the
highly successful 199596 IMA program on materials science)
we were naturally led to the investigation of a class of free
boundary models of biological processes. More precisely, we
undertook a study of some simplified models that have been proposed
to understand the basic mechanisms and the possible control
of tumor growth. Our initial contribution~[Friedman and Reitich,
1998b] consisted of a detailed analysis of radially symmetric
models applicable, for instance, to the socalled "multicellular
spheroids." Our results include the nonlinear asymptotic
stability of steady states within the class of radial solutions.
Stability results are of crucial importance, as they can be
directly correlated to a tumor's ability for local invasion
of surrounding tissue and subsequent metastasis. A true understanding
of stability diagrams, however, demands a thorough description
of possible equilibrium configurations. This, in turn, motivated
our most recent work~[Friedman and Reitich, 1998a] where we
established, to our knowledge for the first time, the existence
of nonradial steady states. Our current efforts are
devoted to the analysis of the stability properties of these
newly found equilibria, which will have obvious implications
in our concurrent search, jointly with J. Lowengrub, of effective
algorithms for the numerical simulation of the growth process.
Regarding the educational activities at IMA we organized, together
with F. Santosa, a workshop for graduate students on Mathematical
Modeling in Industry which was held from July 22 to July
31, 1998. The workshop, the fourth one convened at IMA, brought
together 34 mathematics students from graduate programs across
the country for an intensive 10day modeling experience associated
with industrial problems. The students were divided into six
teams, each working under the guidance of an experienced industrial
researcher who was asked to pose a realworld problem that their
companies need to resolve. As we expected (and, in reality,
as we desired) the problems that were proposed to the students
were not the neat, welldefined academic exercises found in
classrooms, but rather they consisted of stimulating openended
industrial pursuits. In most cases, the problems required new
insight for their formulation and solution. The students spent
ten days working on the problems and were asked to present their
results orally on the last day of the workshop. In addition,
the teams prepared written reports which we have collected in~[Reitich
and Santosa, 1998].
Finally, we have also kept heavily involved in technology transfer
activities at IMA. Indeed, over the last few months we have
initiated collaborative projects with researchers at the Honeywell
Technology Center, Honeywell Inc. (Minneapolis, MN), and at
the MR Head Design division, Seagate Recording Heads (Minneapolis,
MN). The Honeywell project relates to signallaunching onto
multimode optical waveguides. The objective is to design an
effective numerical tool for the prediction of modal energy
distribution upon the guide's illumination. In addition to the
difficulties posed by the possible existence of a large number
of guided modes, the problem can be compounded by manufacturing
imperfections that result in perturbed material or geometrical
parameters. We expect that our recent AFOSR supported work on
analytic continuation methods for problems of wave propagation
will have a bearing on the treatment of this latter problem.
On the other hand, our experience in materials science (and,
particularly, in magnetic composites~[Reitich and Simon, 1997]),
should prove valuable in our joint venture with Seagate. The
goal there is to design models and numerical algorithms that
will aid in the design of the read and write heads within their
hard disk drives. The main mathematical issues to be resolved
relate to nonlinear macro and micromagnetics models and calculations.
Relevant
Publications
 [Friedman
and Reitich, 1998a] A. Friedman and F. Reitich, On the
existence of spatially patterned dormant malignancies in a
model for the growth of nonnecrotic vascular tumors,
submitted.
 [Friedman
and Reitich, 1998b] A. Friedman and F. Reitich, Analysis
of a mathematical model for the growth of tumors, J. Math.
Biol., to appear.
 [Friedman
and Reitich, 1998c]} A. Friedman and F. Reitich, Asymptotic
behavior of solutions of coagulationfragmentation models,
Indiana Univ. Math. J., to appear.
 [Reitich
and Santosa, 1998], F. Reitich and F. Santosa, Mathematical
modeling in industry: IMA summer program for graduate students,
July 2231, 1998, IMA Preprint #1589, October 1998.
 [Reitich
and Simon, 1997], F. Reitich and T. Simon, Modeling and
computation of the overall magnetic properties of magnetorheological
fluids , Proc. of the 36th IEEE Conference on Decision
and Control (1997).
Frances K. Skinner
of the Playfair Neuroscience Unit The Toronto Hospital,
Western Division provides the following impression after
his visit:
I would like to express my sincere thanks for the invitation
to attend the & Computational Neuroscience& workshop
(Jan 1423) in IMA's annual program on "Emerging Applications
of Dynamical Systems." Thank you for the support and
arrangements done on my behalf.
I came home mentally exhausted, but exhilarated at having
learnt so much in a short space of time. It was certainly
a pleasure to have had this opportunity to learn and interact
in this exciting field. I sincerely hope that such a workshop
can be repeated sometime in the future.
J. Leo van Hemmen
of Physik Department, TUMuenchen expresses the following:
I would like to express my sincere gratitude to the IMA
for such a wonderful meeting on Computational Neuroscience.
It is also a great pleasure to me to mention the expert
help of the IMA staff. The meeting was important, and timely,
in that people could meet and discuss in such a stimulating
atmosphere. The more so since the mathematics of neuronal
information processing and its modeling, exciting  and
florishing  as they may be, are still in their infancy.
In short, the meeting was extremely fruitful and stimulating
to me  in fact, to nearly all the participants.
Thanks to all of you!
Steven H. Strogatz
of Cornell University, Theoretical and Applied Mechanics
was a participant in the IMA Workshop on "Computational
Neuroscience" held on January 1423, 1998. He was also
invited to be in residence for spring quarter 1998, Symmetry
and Bifurcation. Below is his report:
I enjoyed the hospitality of the IMA for three months, from
April 1, 1998 to June 30, 1998, as part of the yearlong
program in Emerging Applications of Dynamical Systems. This
was an incredible three months for me  I had no idea the
IMA was so wonderful. I am now one of its biggest cheerleaders.
The administration of the IMA is smooth and seamless, making
the lives of the mathematicians as pleasant as possible.
And the mathematical program was superb, both in terms of
subjects and the caliber of visitors in attendance.
First, let me summarize the collegial aspects that were
such an important part of my time at the IMA. Almost every
day I had stimulating discussions over lunch or dinner with
the postdocs (Warren Weckesser, Kathleen Rogers, Shinya
Watanabe, et al.) and some of the other longterm visitors
(Dwight Barkley, Laurette Tuckerman, John Guckenheimer,
and David Chillingworth were the most frequent companions.)
Even breakfast was a special occasion. Several IMA visitors
stayed at The Wales House, and we got to know each other
in that delightful setting. In particular, for the first
month of my stay, I had the pleasure of eating breakfast
every day with Jurgen Moser, one of the greatest mathematicians
of this century. I'll also have fond memories of breakfast
conversations with Bill Langford, Edgar Knobloch, Ian Melbourne,
Claudia Wulff, and many others too numerous to list.
When I wasn't eating, I worked on the following mathematical
projects:
Laser
dynamics:
My time at the IMA allowed me to work intensively (via email)
with my graduate student Stephen Yeung, enabling us to complete
a paper about the nonlinear dynamics of a laser [1]. Some
geometric aspects of that problem had puzzled us, but with
the help of IMA postdoc Mark Johnson and his visualization
software, we finally began to understand the intricate threedimensional
phase portrait underlying the laser system. IMA postdoc
Ricardo Oliva was also a helpful partner in this project.
I also benefited from conversations with Rachel Kuske, a
professor in the industrial part of Minnesota's math department,
and an expert on laser dynamics. Finally, the IMA provided
great hospitality to my visitors Raj Roy, Henry Abarbanel,
and their students and postdocs  we are all part of an
NSF sponsored collaboration on synchronization and communication
in nonlinear optical systems, and we held one of our quarterly
meetings during a weekend at the IMA.
Time
delay in coupled oscillators:
A second paper with Stephen Yeung [2] deals with the effects
of time delayed coupling on the collective behavior of
the Kuramoto model, a classic model in nonlinear dynamics.
We finished this paper while I was at the IMA, and we also
got some ideas for a new direction on the problem, thanks
to a penetrating remark made by Kurt Wiesenfeld, a visitor
to the IMA during the Pattern Formation workshop. Kurt suggested
his idea during a brainstorming session  just the sort
of informal discussion that happens all the time at the
IMA, and that is so important to mathematical progress.
Smallworld
networks:
In early June, much of my time was spent dealing with the
media excitement about my Nature paper with Duncan Watts
on smallworld networks [3]. In a span of just a few days,
we were interviewed by several major newspapers, magazines,
and radio shows. Articles about our work appeared in the
The New York Times, Washington Post, UPI, Reuters, Business
Week, Science News, and New Scientist, and are scheduled
to appear in Physics Today (September issue), Discover Magazine
(December) and Popular Science. We've also been interviewed
on TV by CBS News and the BBC, and on radio by National
Public Radio (Morning Edition), Voice of America, German
Nationwide Radio, and the BBC. ( A list of web links to
some of these articles appears at the end of this report.)
The IMA graciously took messages as needed, and allowed
me to make international phone calls to these reporters.
Reviews:
Toward the end of my stay, I wrote a News and Views article
for Nature magazine [4] and a book review for Computers
in Physics [5].
Workshops
and seminars:
I also suggested and participated in an informal seminar
on bifurcations with symmetry, led by Warren Weckesser.
This seminar helped several of us beginners come up to speed
with the symmetry aspects of the Pattern Formation workshop.
Other workshops in which I participated were Pattern Formation
(coorganizer), Animal Locomotion (participant), Dynamical
Systems Methods in Oceanography (participant), and Control
theory (participant).
References:
[1] M. K. Stephen Yeung and Steven H. Strogatz, "Nonlinear
dynamics of a solidstate laser with injection," Physical
Review E (in press, October 1998).
[2] M. K. Stephen Yeung and Steven H. Strogatz, "Time
delay in the Kuramoto model of coupled oscillators,"
submitted to Physical Review Letters, July 1998.
[3] D. J. Watts and S. H. Strogatz, "Collective dynamics
of 'smallworld' networks," Nature 393, pp.440442
(1998).
[4] Steven H. Strogatz, "Nonlinear dynamics: Death
by delay," Nature 394, pp. 316317 (1998) (invited
News and Views article).
[5] Steven H. Strogatz, Book Review of "The Genesis
of Simulation in Dynamics," Computers in Physics (in
press).
Giovanni Zanzotto of
CNRUniversita di Padova, Dipartimento di Metodi e Modelli
Matematici submits the following report:
I have participated to the IMA program on `Symmetry breaking
and pattern formation.' Mostly, during my visit I have studied,
with Prof. L. Truskinovsky (Aero Dept., U of MN), the energy
landscape of a class of crystalline materials whose solid
phases have progressively reduced symmetries and a corresponding
sequence of spontaneous symmetry breakdowns. While the project
did not originate at the IMA, I developed it a great deal
during my 1998 visit; also, during that time I did benefit
from some interaction with such renown experts in this field
as M. Golubitsky and I. Stewart, who had organized the IMA
program.
In detail, my research with Prof. Truskinovsky focused on
the description of the temperaturedependent strain energy
function for a crystal exhibiting tetragonalorthorhombicmonoclinic
('tom') martensitic transformations. A very wellknown
material undergoing these phase transitions is for instance
zirconia (ZrO2), which is the main toughening agent in transformationtoughened
ceramics. In the applications (for instance in turbine blades)
the 'active' zirconia inclusions in an inert ceramic matrix
are used to control and enhance the otherwise low ductility
of the ceramic composites.
Our main point of interest was the description of an elastic
crystal exhibiting a tom triple point in its pressuretemperature
phase diagram. As is well known, the variety of available
microstructures increases with the number of coexisting
phases, so one expects that in the vicinity of a triple
point the number of (meta)stable microstructures will also
be the highest a desired effect for improving the performance
of active materials. On this topic we are finishing a paper
titled 'Elastic crystals with a triple point', which later
this Fall will be submitted to the Journal of the Mechanics
and Physics of Solids (the IMA will be acknowledged in the
paper).
In this paper we identified the order parameters and derived
an appropriate `minimal' Landautype energy for the tom
crystals, as the lowestorder polynomial in the strain variables
exhibiting the complete sets of minimizers with the desired
symmetries. This allowed us to study various features of
the triple point and of the nearby region in the phase diagram,
where three distinct sets of energy wells with different
symmetries (parent phase and productphase variants) coexist.
Our energy function is suggested in part by the analysis
of a simple discrete mechanical system with four particles
connected by six LennardJones springs, which shows instabilities
and bifurcations analogous to those characteristic of the
tom crystals.
To our knowledge, ours is the first analysis of a triple
point from the perspective of nonlinear elasticity theory,
which presents the energy function exhibiting the complete
set of the relevant (local and absolute) minimizers. What
we learn from the analysis of the tom crystals is rather
general, and can be largely transferred to other elastic
crystals with nonvariant points in their phase diagram.
Furthermore, our method for writing the constitutive function
suggests a general procedure for constructing energy functions
also for any other type of martensitic symmetrybreaking
transformations involving solid phases with a progressive
reduction of symmetry.
There are several mid and longterm goals that we plan
to pursue in the (near) future, because the results obtained
so far clarify the status of the triple and other nonvariant
points in the phase diagrams of crystalline materials, and
pose several questions which we are currently investigating.
One question can be phrased as follows: How many (meta)stable
phases can be observed for a material in the pressuretemperature
region near a triple point? We have reasons to think that
it is possible to have more than three phases coexisting
in the vicinity of a triple point so that the number of
coexisting phases is larger than what is predicted by the
celebrated Gibbs phase rule.
Since we take metastable states into account (relative energy
minimizers), this observation is actually not in contradiction
with Gibbs' results, but rather represents a nontrivial
extension that may lead to a better understanding of a variety
of processes that take place in the vicinity of the triple
or other nonvariant points (this is a significant issue,
for instance, in the geological and mineralogical applications).
A natural extension of this line of research is the modeling
of materials with more than three phases (numerous materials
are known to exhibit several stable phases with distinct
symmetries). We have written a prototypical energy function
exhibiting up to six types of relative minimizers with distinct
symmetries (cubic, tetragonal, rhombohedral, orthorhombic,
monoclinic, and triclinic) and we plan to investigate this
energy function, studying in particular the pressuretemperature
ranges in which there is coexistence of many (meta)stable
phases. We plan to compare the prediction of our model with
existing experiment.
This research emphasizes the importance of studying specific
regions in the phase diagrams characterized by a particular
abundance of coexisting energy minimizers (relative or absolute).
This is of interest also to materials science because in
these regions an 'active' material may exhibit a dramatically
increased ability to form equilibrium microstructures in
response to the imposed loads, displacements, magnetic fields,
etc., and hence display an improved macroscopic behavior.
One of our future goals is obtaining criteria for the design
of the new active materials whose phase diagrams exhibit
triple points in desirable positions, for instance at pressures
and temperatures closer to room conditions than those at
which nonvariant points are usually found. One main question
regards the methods in which a triple point can be 'moved
around' in the phase diagram. Two main ways can be envisaged
to do so: (a) by selecting specific compositions for suitable
alloys, and (b) by placing transforming inclusions into
elastic matrices which may stabilize otherwise unstable
phases, or by creating thin films where the stabilizing
surface effects are essential.
4. POSTDOCS
Miaohua Jiang currently
affiliated with Wake Forest University, Department of Mathematics
and Computer Science has served as an IMA postdoc during
the 199798 year on " Emerging Applications of Dynamical
Systems." He reports:
My year at the Institute for Mathematics and its Applications
participating the program Emerging Applications of Dynamical
Systems has been an exciting and productive one. I enjoyed
the wellorganized workshops including those special ones:
seminars presented by people from industry; workshop on
NSF new programs; and the summer programs. Besides the educational
benefit of the program  that will be seen in the years
to come, the program provided me an opportunity to work
closely with many program participants to complete several
of my research projects and also initiated many new directions
in my research in applications of dynamical systems. Next
is a list of work I have done during the program year.

Met with Pesin during September workshops on dynamics
of large systems. Finalized our joint work on the existence,
uniqueness, and finite dimensional approximation for SRBmeasures
of hyperbolic lattice systems. The article was accepted
by Comm. Math. Phy. and is published in 1998.

Completed my article on the Entropy Formula for SRBmeasures
which extends the results of the joint work with Pesin.
The article was accepted for publication by the Journal
of Statistical Physics and will be published in 1999.

With the help of the computing facility and personnel
at IMA, I finished research on the dynamics of spatial
averages of quadratic maps with Monte Carlo methods. The
results was submitted to Physics D and the paper is currently
under revision.

Under the direction of John Guckenheimer, I studied the
dynamics and bifurcations of a three dimensional system
modeling thermo instability. I showed that contrary to
common believes, the system does not possess homoclinic
orbits. The technical report was issued as a preprint
of IMA.

Initiated a joint work with Rafael de la Llave on the
smooth dependence of SRBmeasures on the hyperbolic lattice
systems (A paper was finished in April 1999 and submitted
to Comm. Math. Phy.. It was also issued as a preprint
of IMA.)
References
[1] M. Jiang and Y. B. Pesin 1998 Equilibrium Measures for
Coupled Map Lattices: Existence, Uniqueness, and FiniteDimensional
Approximation Commun. Math. Phys. 193, 675711 (also
IMA Preprint series 1525)
[2] M. Jiang 1999 The entropy formula for SRBmeasures of
lattice dynamical systems, to appear, J. Stat. Phys.
[3] M. Jiang, QuasiMonte Carlo studies of spatial averages
of quadratic maps, 1998 17 + 16 Illust., IMA Preprint series
1584, submitted.
[4] M.Jiang, Dynamics and bifurcations of a 3D system modeling
thermal instability, 1998 113 + 6 Illust., IMA Preprint
series 1583, submitted.
[5] M. Jiang and R. de la LLave, Smooth Dependence of Thermodynamic
Limits of SRBmeasures, 134, 1999, IMA Preprint series
1615, submitted.
Mark E. Johnson, who
received his PhD from Princeton University's Program in
Applied and Computational Mathematics, can now be found
in the webspace as a Technical Yahoo at Yahoo!, Inc. His
statement follows:
During my oneyear position as a postdoc at the IMA's year
on Emerging Application of Dynamical Systems, I had the
privilege to interact with an excellent crosssection of
the dynamical systems community in an environment ideal
for such purpose.
Some of the highlights of my year at the IMA are as follows:
1. During one of the initial workshop's Software Demonstration
sessions, I performed a realtime demonstration of SCIGMA,
a software application which was developed in collaboration
with Michael S. Jolly (one of the program's yearlong visitors)
and Yannis G. Kevrekidis (one of the program's organizers
and my PhD advisor). SCIGMA is a interactive graphics tool
developed for Silicon Graphics workstations. By computing
and visualizing collections of invariant objects with SCIGMA
in the phase space of a dynamical system, one can make powerful
observations about the geometry of complex objects within
this space.
The software demonstration was used to illustrate to the
audience some of the powers of such a tool. This demonstration
also led to the organization of an exciting evening session,
held at the Geometry Center, involving the comparison of
several similar algorithms and applications developed by
program participants Michael Dellnitz, Hinke Osinga &
Bernd Krauskopf, John Guckenheimer, and Eusebius Doedel.
In addition to these people, there were several other visitors
in attendance (Bernold Fiedler, Jim Yorke, and Marty Golubitsky
to name a few).
Such an interaction would normally be considered rare, but
in the IMA's environment, is considered typical.
2. Ongoing research with Mike Jolly continued during our
overlapping time at the IMA. We continued work, also in
collaboration with Yannis Kevrekidis, on using the SCIGMA
software application to uncover interesting global bifurcation
phenomena found to occur in a threedimensional approximate
inertial form for the onedimensional periodic KuramotoSivashinky
equation. These observations were made by computing and
observing the interaction of collections of twodimensional
stable and unstable invariant manifolds of steady states
and limit cycles. The observed phenomena has been described
in a recently submitted paper called "The Oseberg transition:
visualization of global bifurcations for the KuramotoSivashinsky
equation". When this work was still in progress, I
delivered a Postdoc Seminar in April, 1998, called "Interacting
twodimensional manifolds and global bifurcations for the
KuramotoSivashinsky Equation."
3. The longtime dynamics of the KuramotoSivashinsky equation
has been studied extensively under the assumption that the
equation model phenomena on a homogenous medium. A question
that was studied with Edriss Titi, several times a visitor
to the IMA during the year, and Yannis Kevrekidis, was how
spatial perturbations might affect the longtime dynamics
(such as the existence and radius of its absorbing ball,
dimension of the attractor, etc.) of this model equation.
The results of our efforts appear as a chapter in my thesis.
In addition to these interactions, I enjoyed numerous interactions
with George Sell, Steve Strogatz, Herb Keller, Ciprian Foias,
Dwight Barkley, and, of course, the other postdocs and visitors
throughout the year. None of this would have been possible
without the organization and hospitality of the IMA's staff.
Kurt Lust of United
Technologies Research Center is one of the IMA Postdoctoral
Members in Industrial Mathematics. He has the following
report:
From September 1997 till June 30, I was a postdoctoral member
for the 199798 year on Emerging Applications of Dynamical
Systems. In July, I became a postdoctoral member in industrial
mathematics, working with the Dynamic Modeling and Analysis
group at the United Technologies Research Center in East
Hartford, Connecticut. Research at the IMA My dissertation
research concentrated on numerical methods for bifurcation
analysis, in particular for large scale problems with lowdimensional
dynamics. Initially, I continued work in this direction.
During my Ph.D. research, I developed methods based on single
and multiple shooting for bifurcation analysis of largescale
problems. I made some improvements to my single shooting
based code, and development of the multiple shooting based
code continued. I also developed a method to compute the
Floquet multipliers in multiple shooting and some collocation
codes with very high accuracy. This method is based on a
matrix decomposition (the periodic Schur decomposition)
developed at the IMA during the year on Applied Linear Algebra
(19911992) by G. Golub, P. Van Dooren and A. Bojanczyk.
During my stay at the IMA, I made some refinements to my
implementation. I also set up a test case together with
E. Doedel, using matrices generated by AUTO97. The results
were very good. J. Guckenheimer and S. Watanabe have also
used the code in some of their experiments with a novel
method for computing periodic solutions developed by J.
Guckenheimer.
I prepared an article on the convergence behavior of my
single shooting based methods that will appear in the proceedings
of the workshop on Large Scale Dynamical Systems, held at
the IMA in September 1997.
Another paper, which I started writing at the K.U.Leuven,
was revised and sent back to the IMA Journal on Numerical
Analysis. Some time was also spent in investigating future
development, e.g., extension towards collocation, both for
partial differential equations and delay differential equations,
and implementation on parallel computers.
At the end of February, I had a job interview at the United
Technologies Research Center. This company is very interested
in using dynamical systems techniques to enhance the performance
of their products. Though the result of the job interview
was positive, I decided that I really wanted to spend a
year at Cornell University with J. Guckenheimer. A project
was set up together with United Technologies, which lead
to the change to industrial postdoc status during July and
August. During my stay at United Technologies, I spent most
of my time working on an alternative technique for bifurcation
analysis of largescale problems. Instead of doing a bifurcation
analysis of the largescale problem, a lowdimensional model
with similar behavior is constructed and analyzed. The goal
was to build a model for the large coherent structures observed
in fluid flow in a diffuser for varying angles of the diffuser
and at a fairly high Reynolds number. The model must be
able to capture behavior at varying angles since one goal
was to study bifurcations as the angle of the diffuser varies.
The approach chosen was to gather data from simulations
using a CFD code, analyze the data using the proper orthogonal
decomposition technique, and then build a model by projecting
on the POD modes. Because of the complex structure of the
equations (our largescale model was the NavierStokes equations
for a compressible fluid) and the varying spatial domain,
we have chosen not to do a Galerkin projection of the system
of equations. This would not lead to a cheap in our case,
and is very hard to do. Instead, we did a projection of
the system of ordinary differential equations defined by
the CFD code. This procedure is implemented as a layer around
the existing CFD code and grid generation code. This method
is not cheap either, but is much easier to program and easier
to extend to other equations. The model was implemented
as a Matlab MEX routine written in Fortran 77 (the language
of the CFD code we used). Some first results with the model
were obtained by the end of August. Simultaneously, other
researchers at UTRC, together with Y. Keverekidis, P. Holmes
and a Ph.D. student at Princeton, are investigating other
types of modeling such as blackbox modeling using artificial
neural networks and a traditional Galerkin projection of
the NavierStokes equations for incompressible flow. During
my stay at UTRC, I've also optimized their CFD code for
their workstations and made various improvements to enhance
the robustness of the code. We have also discussed the use
of tools for bifurcation analysis of largescale problems
for their problems.
It was very interesting to learn how research is done in
an industrial environment. A good understanding of how companies
work is essential in setting up successful collaborations
with industry. I am convinced that the experiences which
I gained during the summer will be very useful in my later
career, whether in academics or in an industrial or government
research lab, and I feel that more researchers in academics
should spend some time in such an environment. Other aspects
I joined all the workshops held at the IMA during the period
September to June and went to most of the talks in those
workshops. From the core program workshops, the fall workshops
were most interesting for me though I also learned many
interesting ideas from the spring workshops. In the winter
workshops, I learned about an other application area I was
not familiar with before my stay at the IMA. I also very
much enjoyed the workshop on "Algorithmic Methods for
Semiconductor Circuitry" held in the fall and the workshop
on Dynamical Systems in Oceanography that was added to the
program in the spring. The workshop on Algorithmic Methods
for Semiconductor Circuitry was a clear proof of the lack
of communication between different research areas: in several
talks, primitive numerical methods were presented that have
been known for a long time in other areas and have reached
a high level of sophistication. This clearly demonstrates
the importance of bringing together people working in various,
strongly or only vaguely related, areas. A successful example
of such a workshop is the workshop on Animal Locomotion
and Robotics, held at the IMA in June.
I also had many interesting conversations with visitors
at the IMA. I had several discussions with H. Keller during
the fall. The algorithms which I developed are based on
his work with various graduate students, and it was interesting
to learn about new developments at Caltech. I also had interesting
discussions with some of the other visitors during the fall,
including W.J. Beyn, W. Govaerts, M. Friedman and L. Tuckerman.
I had several discussions with Y. Kevrekidis during his
visits at the IMA and have tried to set up some collaboration.
Due to lack of time, much of that work is done by a Ph.D.
student in Leuven. Y. Kevrekidis is also cooperating in
the project on lowdimensional modeling using PODs. I hope
I can collaborate more intensely with him at Cornell University.
I also got in contact with D. Barkley of the University
of Warwick in the U.K., and it is very likely that I will
move to Warwick after my stay at Cornell University if I
decide to stay in academics. Conclusion The environment
of the IMA had a very broadening effect on my research.
I've laid several contacts that will hopefully last long
and determine my further career. The collaboration with
United Technologies was also very successful and will be
continued during my stay at Cornell. Together with United
Technologies, I will continue to work on the modeling of
the diffuser and start a new project on modeling of combustion
instabilities, which is a new area for me. Combustion is
an area of quickly growing importance and a good choice
to further broaden my research. My skills on analysis of
dynamical systems will be very useful in both projects.
I am convinced that the year at the IMA helped me a lot
in getting ready for a tenure track position or a senior
research position in a research laboratory. It will definitely
have a large effect on my further career in research.
RolfMartin Mantel
is one of the 199798 regular postdocs. He writes:
I spent a productive year at the IMA in the programme on
Applications of Dynamical Systems. I actively participated
in most workshops, especially by giving a computer demonstration
on spiral waves at the Workshop on Computational Neuroscience.
I also participated in the postdoc seminars, where I presented
my PhD research, and in informal seminars on continuous
spectra in fall and on bifurcations with symmetry in spring.
I prepared one paper of my PhD thesis for publication as
an IMA preprint. It was cowritten with Dwight Barkley of
the University of Warwick, who was also a yearlong visitor
at the IMA. Its title is "Parametric forcing of scrollwave
patterns in threedimensional excitable media."
During my year here, I entered a close collaboration with
Bernold Fiedler, who was a visitor in fall and did several
subsequent short visits. As a result, we have a multimedia
paper in preparation with the working title "Crossover
Collisions of Scroll Wave Filaments."
In this paper, we take Dwight Barkley's computer simulation
"EZSCROLL" that simulates scroll waves in three
space dimensions and adapt it with a new filament detection
algorithm that also detects collision of filaments. We show
various examples of scroll wave collisions, and we prove
analytically that there is only one generic way scroll waves
can collide. My contribution to the paper was concentrated
on the numerical side. I pursued some research on the meander
to hypermeander bifurcations of spiral waves in two dimensions.
I was able to detect the onset of several meander frequencies
but did not find any unbounded motion at high spatial resolutions.
I learned that calculating the locus of the bifurcations
leading to the added meander frequencies is a very challenging
numerical problem.
The neuroscience workshop encouraged me to look for "lurching"
spirals in two space dimensions. I found out that adapting
PDE software from diffusion coupling to synnaptic coupling
is a major task that challenges several of the core assumptions
of the "EZSPIRAL" software.
Out of the contacts with visitors I made, I was offered
a job by Klaus Boehmer, and I aim to do future collaboration
with Michael Dellnitz. I was also glad to meet several people
working in topics closely related to mine, especially Claudia
Wulff and Bernd Sandstede.
Marina Osipchuk of
Honeywell Technology Center (HTC) is one of the 199798
Industrial Postdoctoral Members. She reports:
As a participant of the Industrial Program affiliated with
the Honeywell Technology Center (HTC) I enjoyed the combination
of the industryrelated research and collaboration with
the IMA participants for the program year. In the course
of the project at the HTC we developed a lowcost accurate
contact establishment and attitude determination strategy
for a constellation of low Earth orbit satellites. While
working on my industrial project I benefited from discussions
with the organizers of the program: Profs. Fadil Santosa,
Blaise Morton, Avner Friedman, Walter Littman, Rachel Kuske
as well as the other industrial postdocs: Gilberto Lopez
and Shari Moscow.
Participation in the annual program on "Emerging Applications
of Dynamical Systems" expanded my knowledge in latest
theoretical issues in dynamical systems. The scope of application
areas presented at the workshops fascinated me. I enjoyed
the opportunity to work closely with the participants of
the workshops (Edriss Titi, Yannis Kevrekidis, Stas Shvartsman)
on control of reactiondiffusion systems.
I also participated in an informal seminar on bifurcations
with symmetry organized by Warren Weckesser. The seminar
study served as a great introduction for the forthcoming
workshop on Pattern Formation. In addition I participated
in a weekly postdoc seminar and gave an overview talk on
control of nonlinear systems as an introduction to the workshop
on Nonlinear Identification and Control.
In addition, I revised a paper on the research that I started
at the University of California, Irvine. The paper "Geometry
and Inverse Optimality in Global Attitude Stabilization"
was published in Journal of Guidance, Control and Dynamics.
Research
Attitude determination in satellite constellation
The "fiberlike" quality along with globally provided,
spacebased communication are the distinctive features of
the broadband lowEarth orbit (LEO) satellite system Teledesic.
The high speed, high quality of data transmission is achieved
via intersatellite communication (each satellite has laser
links to eight adjacent satellites). To handle the system
networking the satellites have to be positioned and oriented
much more precisely than satellites that communicate only
with the ground. In addition, to facilitate satellite crosslinks
the satellites will have onboard computers, thus increasing
the cost of the system and power consumption in the orbit.
Our goal was to develop navigation concepts that employ
the existing communication system and thus eliminate the
additional cost of conventional sensors.
The developed attitude determination strategy employs maximum
probability search to establish contacts between nonoriented
satellites in the groups and updates the estimate of the
satellite attitude. Once the satellites are linked within
the group the accuracy in attitude determination is limited
only by hardware resolution. Efficiency in contact establishment
is guaranteed while using the suggested nonlinear programming
search. Finally, the described attitude determination method
exploits existing communication hardware thus eliminating
additional cost for conventional sensors.
Nonlinear control of dissipative evolutionary equations
The dynamics of the evolutionary equation is governed by
a PDE with infinitedimensional state of the corresponding
control problem. On the other hand, the implementation issues
such as discrete location of actuators, limited memory capacity
and finite precision of computers call for a finitedimensional
control.
A nonlinear model reduction method of the discretized PDE
was employed to approximate the longterm behavior of the
PDE dynamics by a dynamical system of finite, small dimension.
Closing the obtained system with a linear controller effectively
stabilizes the PDE truncation locally; this does not, however,
exploit the fully nonlinear reduced model. On the other
hand, nonlinear, feedback linearizing control results in
large control spillover to the residual modes. We design
an inverse optimal control law for the reduced system. The
stability and robustness of the full closedloop system
with respect to fast unmodeled dynamics was analyzed using
singular perturbation and bifurcation techniques. Simulation
results demonstrate the performance advantages of the inverse
optimal control law over the linear quadratic regulator
in stabilizing models derived from our illustrative example.
Kathleen A. Rogers
of the University of Maryland is on her 1st year as an IMA
Postdoc. She writes:
Workshops and Seminars
As a Postdoctoral Member of the IMA during the theme year
Emerging Applications of Dynamical Systems, my main responsibilities
included attending workshops, seminars and tutorials. This
provided a unique opportunity for me to gain a broad perspective
of the latest theoretical and numerical issues in dynamical
systems as well as to become acquainted with some new application
areas. After completing my thesis last year, I felt that
I was lacking a more global view of current research in
applied mathematics, and dynamical systems in particular.
I took this opportunity to expand my knowledge in several
areas of dynamical systems such as geometric singular perturbation
theory, continuous spectrum of operators, bifurcation in
the presence of symmetry, and pattern formation in reactiondiffusion
systems, just to name a few. I was also intrigued by the
variety of application areas that were presented throughout
the year. Workshops which focused on neurobiology, cardiac
rhythms, and calcium dynamics provided an in depth look
at applications in which mathematics is just starting to
make an impact. During the course of the year, I also enjoyed
hearing about the other applications of various techniques
and methods in dynamical systems. For instance, one invited
speaker focused on the role differentialalgebraic equations
play in modeling chemical reactions and another focused
on using reactiondiffusion equations to model the life
cycle of mold spores.
In addition to the scheduled workshops and seminars, I participated
in two informal weekly seminars. In the fall, Bernold Feidler
organized the `working' seminar on continuous spectra. The
seminar began with understanding the definitions of the
various kinds of spectra that operators can have and continued
onto discussions about algorithms for numerically calculating
continuous spectra. During the spring quarter, Warren Weckesser
organized a `working' seminar on bifurcations in the presence
of symmetry. This seminar consisted of reading some elementary
papers on bifurcations in the presence of symmetry in order
to become familiar with the vocabulary. There were also
presentations of `work in progress' on which we tried to
offer suggestions.
Talks
In addition to participating in the workshops and seminars
associated with dynamical systems, postdoctoral members
are expected to organize, attend and speak at the weekly
`Postdoc Seminar'. I volunteered to organize the postdoc
seminar for the winter session, in which the workshops focused
on biological application areas. Since these were new areas
for most of the postdoctoral members, I tried to incorporate
an introduction to the areas as part of the regular postdoc
seminars. As with all the postdocs, I gave a talk in the
postdoc seminar. I was also invited to speak at the Applied
Math Seminar at the University of Arizona.
Research
My research accomplishments for this year encompassed three
separate projects. The first project involved writing papers
on my dissertation research, stability of twisted elastic
rods as a model for supercoiling in DNA minicircles. The
second project was an industrial problem presented to the
IMA by General Motors involving welding and clamping of
beams. The third project investigated a system of four ordinary
differential equations that serve as an idealized model
of two reciprocally inhibitory neurons.
DNA:
A twisted elastic rod is widely accepted to be a qualitative
model of supercoiled DNA. Mathematically, a twisted elastic
rod is represented by an isoperimetrically constrained calculus
of variations problem. That is, the equilibria of the rod
exactly correspond to critical points of a certain functional
subject to integral constraints. Similarly, critical points
which correspond to constrained minima are said to be stable
equilibria. My thesis comprises a series of practical tests
which determine which critical points correspond to constrained
minima, or equivalently, which equilibria are stable. My
research goals in this particular area were to complete
papers that were based on my thesis research. During this
year, one of the papers
R.S. Manning, K.A. Rogers, & J.H. Maddocks, Isoperimetric
Conjugate Points with Application to the Stability of DNA
Minicircles
was accepted and will appear in the Proceedings of the Royal
Society of London: Mathematical, Physical and Engineering
Sciences. Additionally, a paper with Leon Greenberg and
John Maddocks was written and submitted:
L. Greenberg, J.H. Maddocks, & K.A. Rogers, The
Bordered Operator and the Index of a Constrained Critical
Point.
During a trip to EPFL in Lausanne Switzerland to visit my
advisor, the results for the final paper from my dissertation
were strengthened and generalized to include stability exchange
results at nonsimple folds as well as simple folds. These
generalizations required a significant rewrite of the paper
K.A. Rogers & J.H. Maddocks, Distinguished Bifurcation
Diagrams for Isoperimetric Calculus of Variations Problems
and the Stability of a Twisted Elastic Loop.
This last paper is still in preparation, but should be submitted
shortly.
Welding
and Clamping of Beams: Experiments on shells have demonstrated
that the sequence in which two shells are clamped and welded
affects the final shape of the shells. Such a situation
arises in assembling automobiles. In that setting, the consequences
of different final shapes can be costly if, for instance,
the final shape of the two shells (or automobile parts)
causes the larger structure not to meet required specifications.
In order to understand why this sequence dependence arises
Dr. Danny Baker and Dr. Samuel Marin of General Motors Research
and Development Center, Fadil Santosa, Associate Director
for Industrial Programs at the IMA, and I proposed models
of clamping and welding of beams which demonstrate this
sequence dependence.
Consider the variational formulation of beams with small
initial curvature derived by Washizu (Variational Methods
in Elasticity and Plasticity,Pergamon Press, 1982) in which
the initial shape of each beam will be denoted by f_{1}(s)
and f_{2}(s), the perturbation from the initial
shape in the horizontal direction will be denoted by u_{i}(s)
and the perturbation from the initial shape in the vertical
direction will be denoted by w_{i}(s). The coordinates
of the beams after displacement will be given by x_{i}=s+u_{i}(s)
and y_{i}=f_{isub>i}(s).
The first model which we considered allowed only vertical
displacement of the beam (u_{i}(s)
0, w_i (s)
0), which implies that the clamping mechanism does not allow
`slipping' of the beam as the clamp is closed. Effectively
the clamping mechanism takes a prespecified point on each
beam, s_{1} and s_{2}, and forces it to
a prespecified location y_{0} with a prespecified
slope y_{0}'. The clamping problem is modeled by
two fourthorder ordinary differential equations subject
to left end boundary conditions, continuity conditions at
s_{i}, and the clamping conditions. This model problem
has an analytic solution.
Unlike the clamping procedure, the welding procedure does
not specify the position y_{0} or the orientation
y_{0}' of the beam. The welding conditions specify
that the vertical position and tangent of both beams are
the same and also that the axial force and bending moment
balance at the welding point. This welding procedure is
also modeled by two fourthorder ordinary differential equations
subject to left end boundary conditions, continuity conditions
and the welding conditions. This model also admits an analytic
solution.
If one continues to follow the solution to these models
of clamping and welding through the application of two clamps
and two welds, then one will find no sequence dependence
arises in this model. We then conjectured that the sequence
dependence arises as a result of horizontal displacement
in the clamping procedure. This means that as the clamp
closes each beam is allowed to slide without friction. To
test this theory, we proposed a model which allows both
horizontal and vertical displacements (u_{i}(s)
0, w_{i}(s)
0), where
the clamps and welds are modeled in a manner similar to
the model described above, except of course, it takes into
account the horizontal displacements.
The system of ordinary differential equations including
the welding and clamping constraints that accounts for horizontal
displacement is significantly more complicated than the
simpler model and do not admit an analytic solution. In
order to see the sequence dependence in the model, each
step must be solved numerically. A multiple shooting algorithm
programmed in Matlab is able to compute the solution for
one clamp and the solution for one weld. Unfortunately,
the problems with one weld and one clamp and two clamps
involve two fourthorder equations which are coupled nonlinearly
through clamping conditions, welding conditions and continuity
conditions. This problem cannot be solved by the current
shooting algorithm and is a focus of ongoing research.
Reciprocal
Inhibitory Neurons: Many of the behaviors observed in
the solutions of the Hodgkin and Huxley equations can also
be seen in simpler, yet still biologically reasonable, models.
In particular, simple models of the action potential of
neurons connected by reciprocally inhibited synapses have
been studied to further understand such biological phenomena
as heartbeat, swimming, and feeding. Two identical oscillatory
neurons connected by reciprocally inhibitory synapses will
oscillate exactly out of phase of each other, that is, while
one neuron is active the other is quiescent. John Guckenheimer,
Warren Weckesser and I studied an idealized model of a pair
of reciprocally inhibited neurons in the gastric mill circuit
of a lobster. Our goal is to understand solutions of a set
of four differential equations which model two asymmetric
oscillators in terms of geometric singular perturbation
theory, an effective tool for understanding equations with
multiple time scales. Essentially, singular perturbation
theory pieces together solutions from the fast system and
solutions from the slow system to get a solution of the
singularly perturbed system. Behavior of a singularly perturbed
system consists of motion on the slow manifold (the set
of equilibria of the fast system) and fast jumps between
different parts of the slow manifold. These fast transitions
occur at folds in the slow manifold. A periodic solution
to the system of equations which describes a pair of identical
reciprocally inhibitory neurons can be described in terms
of singular perturbation theory as consisting of two fast
transitions. These fast transitions correspond to one neuron
jumping from an active to a quiescent state and the other
jumping from a quiescent state to an active state.
As we investigated the solution space of the asymmetric
problem, we found many solutions that were qualitatively
similar to the solutions of the symmetric system. We also
found other very different types of behavior. For instance,
in a small parameter range, there exists (at least) two
stable periodic orbits of the full system. Both of these
periodic solutions correspond to more complicated behavior
than the typical reciprocally inhibitory behavior described
above. Instead of the orbit consisting of two fast transitions,
the periodic orbits consist of nine and eleven fast transitions,
respectively, and the behavior of the two neurons can no
longer be classified simply as active or quiescent. The
possible implications of the existence of two stable periodic
solutions as well as the structure of these solutions are
a source of continued research.
In addition to bistability in the system, we also found
canard solutions, that is, solutions in which part of the
orbit occurs on an unstable portion of the slow manifold.
We identified two different types of canard solutions. One
type of canard solution consists of a fast transition to
an unstable part of the slow manifold. In the other type
of canard solution, the orbit continues past a fold in the
slow manifold onto the unstable part of the manifold. For
a small parameter range, the family of canard solutions
is stable. Continuation of the two stable periodic solutions
reveals that the canard solutions persist for a much larger
parameter regime but are unstable.
Tony Shardlow of
OCIAM, Mathematical Institute describes his activities and
achievements during his stay at the IMA as a postdoctoral
member of the Emerging Applications of Dynamical Systems
year.
Long
time approximation of Markov Chains This project
looks at a class of Markov processes arising from stochastic
differential equations and asks to what extent numerical
approximations of these processes can reproduce their long
time behaviours. To gain results in this direction, the
processes are assumed to be geometrically ergodic, which
may be verified in a variety of circumstances and is much
easier to establish than similar notions (hyperbolic attractors,
for example) in deterministic systems. The paper Shardlow
and Stuart (An abstract perturbation theory for Markov chains,
IMA report no. 1563 ) describes how long time averages of
the approximation converge to the true ergodic average of
the underlying process. The approximations must converge
to the underlying process, but it is only necessary to do
so in the sense of distributions on a time interval bounded
away from the origin. This opens the door for example to
weak approximations of PDEs, where error estimates for numerical
methods are generally singular at time t=0.
Motivated by this paper, I developed the results necessary
to apply the above theory to stochastic PDEs. Two results
are needed: (i) finite time convergence for the numerical
method, which I have worked out for finite difference approximations
in (Shardlow, Numerical methods for stochastic PDEs, submitted
Num. Func. Anal. and Opt.) and (ii) geometric ergodicity,
which is discussed in (Shardlow, Geometric Ergodicity for
Stochastic PDEs, to appear Stoch. Anal. App). Further motivated
by the above work, I have started looking into ways of proving
geometric ergodicity when the stochastic term is degenerate.
One class of such problems (hypoelliptic systems) came up
in the IMA workshop on control theory and this may be a
direction for further research.
Stochastic
PDEs for phase transitions This project concerns
stochastic PDEs arising in the modelling of certain growth
processes and are familiar as the CahnHilliard and AllanCahn
equations with an additional randomly fluctuating driving
term. I spent a considerable amount of time reading background
on these equations, leading to a proposal to the EPSRC in
December for a postdoctoral position at the University of
Oxford. There are a number of interesting directions for
this work, related to extending the extensive work on the
CahnHilliard equation to include the stochastic terms.
I'm particularly interested in developing numerical methods
and possibly the use of approximating particle systems.
This proposal was successful and I start studying these
questions in the Oxford Centre for Industrial and Applied
Mathematics (OCIAM) in October.
Whilst I've been at the IMA, my research in this area has
been focused on the 1d AllanCahn equation forced by additive
spacetime white noise and the development of an SDE to
account for the motion of the fronts. Previous work on the
deterministic AllanCahn equation has been successful in
describing the motion of the fronts by an ODE in the relative
positions of the fronts; it is an interesting idea to explore
this idea in the presence of noise. This work involved formal
asymptotics together with thorough numerical experiments,
and is currently being written up (Shardlow, Perturbations
of the AllanCahn equation).
Other
During my stay at the IMA, I have benefited from attending
a large number of seminars in the Dynamical Systems year
over a terrific range of mathematics. Stanford (my previous
institution) is not a centre for dynamical systems and it
has been a breath a fresh air to attend seminars with a
different focus. I presented several talks during the year
including the IMA postdoc seminar twice (Dec 97, Linear
Multistep Methods and Inertial Manifolds; Mar 97, Long time
approximation of SDEs) and the applied math seminar at Wisconsin
(Feb 97, Long time approximation of stochastic PDEs).
Shinya Watanabe currently
affiliated with Ibaraki University, Department of Mathematical
Sciences, has served as an IMA postdoc during the 199798
year on " Emerging Applications of Dynamical Systems."
He reports:
I participated in the IMA year as a postdoctoral fellow
from September, 1997, till June, 1998. This followed my
doctoral program until 1995 and my first postdoctoral appointment
elsewhere during 199597. Since I came to the IMA with some
research experience that had been focused and rather narrow,
my main aim during the IMA year was to broaden my interest
and knowledge in the large field of dynamical systems and
to come to know active researchers in various branches in
the field. With the amazing number of highquality visitors,
seminars, and workshops, the IMA program became an exciting
and fruitful year that would benefit me in my future research.
During the year I attended many talks and was exposed to
many areas that were new to me, from mathematical biology,
locomotion and robotics, to oceanography. There were excellent
tutorials before workshops by such experts as Jim Keener,
Chris Jones, Marty Golubitsky, Joel Keizer, Nancy Kopell,
David Terman, and others. Sometimes postdocs themselves
asked longterm visitors, such as John Guckenheimer and
David Chillingworth, to help us in extra study sessions.
These orientations starting from an elementary level were
indispensable for me as pointers when I tried to follow
discussions during workshops. My knowledge on mathematical
biology, for instance, had been minimal, but, after enduring
the winter quarter that was sometimes overwhelming, I gained
reasonable understanding of the current state of modelling
efforts of neural and cardiac systems. It was also good
to learn what kind of gap there was between mathematicians
and biologists, and how they were trying to close it. I
also learned new techniques in the areas I was more familiar
with, such as geometric singular perturbation theory in
the multiplescale workshop. Reduction of large systems
into smaller degrees of freedom has been one of the areas
of my strong interest, and I came to know different approaches
and styles, e.g., inertial manifolds, numerical decompositions,
and separation of multiple scales.
The IMA policy limiting the number of seminars made room
for interacting with the visitors outside seminar rooms:
in offices, hallways, restaurants, cafes, etc. Because guests
were so knowledgeable that it was rather difficult for me
to have full discussions with them, but I benefitted from
conversations with Anatoly Neishtadt, Eusebius Doedel, Kurt
Lust, John Guckenheimer, Don Aronson, Jim Swift, Greg King,
David Golomb, Richard Haberman, Robert Miura, Steve Strogatz,
just to name a few that come to my mind immediately.
Apart from attending the workshops and seminars, I prepared
articles on the continuing research projects. In October
I finished and submitted a paper with Mauricio Barahona
in Stanford which was published in the following spring
[1]. Also in October, I received a referee report on a short
letter I wrote with my former colleagues and their students
in the Niels Bohr Institute. It was edited and resubmitted
in November, and appeared also in the following spring [2].
A more detailed manuscript was prepared during my stay in
the IMA, and was submitted in the fall of 1998. It appeared
shortly later [3]. During the IMA Pattern Formation Workshop
in the spring, Herre van der Zant visited the IMA as a speaker.
Organizers requested speakers to write a review of the subjects
they presented. I joined him in writing a note on dynamics
in Josephson junction arrays, which will appear as a contribution
to the IMA Proceedings [4].
Since I came to the IMA without a fixed plan after the term,
a portion of my time during the winter was spent on searching
an academic position. Apart from simply stating that I was
offered a satisfying position in the end, I'd like to add
an episode during the search process that demonstrated that
the IMA was "the place to be" during the year
199798. When I went to a university in the U.K. as one
of the five finalists to be interviewed (all five in the
same day), another candidate was a familiar longterm visitor
at the IMA and yet another came for two workshops here.
I was quite proud of the fact that one of "us"
was offered the job there.
Before concluding the note I would like to thank organizers
of the IMA program, the IMA administrative staffs, and the
fellow postdocs for all the support I received.
References
[1] M. Barahona and S. Watanabe
"Rowswitched states in 2D underdamped Josephson junction
arrays" Physical Review B, vol.57, no.17 (1998
May 1) 1089310912.
[2] C. Ellegaard, A. E. Hansen, A. Haaning, K. Hansen, A.
Marcussen, T. Bohr, J. L. Hansen, & S. Watanabe
"Creating corners in kitchen sinks" Nature,
vol.392 (1998 Apr. 23) 767768 (Scientific Correspondence).
[3] C. Ellegaard, A. E. Hansen, A. Haaning, K. Hansen, A.
Marcussen, T. Bohr, J. L. Hansen, & S. Watanabe
"Cover illustration: Polygonal hydraulic jumps"
Nonlinearity, vol.12, no.1 (1999 Jan.) 17.
[4] H. S. J. van der Zant and S. Watanabe,
"Dynamics of kinks and vortices in Josephsonjunction
arrays" to appear in "Pattern Formation in Continuous
and Coupled Systems," IMA Volumes in Mathematics and
its Applications, vol.115, (eds. M.Golubitsky, D.Luss, and
S.H.Strogatz), 283302.

_{}
Warren Weckesser of Rensselaer
Polytechnic Institute is one of the IMA Postdocs. He is on his
1st year of a twoyear term. His report follows:
The year on Emerging Applications of Dynamical Systems was an
exciting one. A large part of my time was spent attending the
IMA workshops, and also several informal seminars that ran during
the year. I learned a great deal about several areas in dynamical
systems, including numerical methods for large scale systems,
multiple time scales, bifurcation with symmetry, nonlinear control
theory and system identification, and biological applications
including calcium dynamics, cardiac dynamics and animal locomotion.
The opportunity to meet with leading researchers in these fields
was invaluable.
Part of my time was spent preparing and submitting papers based
on my Ph.D. thesis. One paper shows that in mechanical systems
with rotational symmetry (under a set of conditions that hold
generically), the relative equilibria sufficiently close to
a stable equilibrium are linearly orbitally stable. I also show
in this paper that the first whirling mode to bifurcate is nonlinearly
orbitally stable, but the standard variational method for proving
nonlinear orbital stability fails in all modes after the first.
Another paper applies these ideas to the hanging chain.
Mark Levi visited the IMA in the fall of 1997, and we had several
discussions on the kinematics of a constant velocity joint.
This led me to begin studying mechanical systems composed of
symmetric rigid bodies coupled with constant velocity joints.
Unlike a universal joint, a constant velocity joint creates
a kinematic constraint that directly couples the angular velocities
of the two rigid bodies. I am investigating the bifurcation
and stability of whirling configurations of chains of coupled
rigid bodies. This work may shed new light on certain gyroscopic
phenomena in spinning beams and related mechanical systems.
I visited Boston University in the fall, and discussed some
of this work with Tasso Kaper, Carson Chow, John Ballieul, and
their students.
In the spring, I began collaborating with John Guckenheimer
and Kathleen Rogers on an intensive study of the rich dynamical
behavior found in a system of two coupled relaxation oscillators.
More specifically, we are considering two nonidentical Van~der~Pollike
oscillators. The coupling is based on reciprocal inhibition,
as occurs in membrane models of neurons. Our model of two coupled
neurons results in a singularly perturbed system of differential
equations, with two fast variables and two slow variables. Our
observations so far include several families of periodic orbits,
a range of parameters for which there are two stable periodic
orbits, families of orbits that exhibit a variety of canards,
and a possible homoclinic bifurcation from a periodic orbit.
One goal of this research is to classify the types of bifurcations
that occur in singularly perturbed systems with more than two
dimensions. Especially important for this work are the methods
of geometric singular perturbation theory. Another important
component of the work so far has been the numerical continuation
of periodic orbits with the software package AUTO. We benefited
from discussions with E.~Doedel, the author of AUTO, during
his visits to the IMA.
In the spring quarter, I was cochair of the IMA Postdoc Seminar.
I also organized (with some initial inspiration from Steve Strogatz)
the "Informal Seminar on Bifurcation with Symmetry,"
a weekly series of seminars and discussions on bifurcation with
symmetry. When a bifurcation problem possesses symmetry, certain
degeneracies occur that make standard bifurcation theory inapplicable.
However, it is possble to exploit the added structure imposed
by the symmetry. There is now an extensive theory and body of
literature on this subject. The goal of the informal seminar
was to learn about this theory, starting with the most elementary
ideas and progressing far enough to be comfortable reading current
literature in the field. Here is partial list of fairly regular
participants.
Postdocs:
Miahua Jiang, Shinya Watanabe, Kathleen Rogers, Rolf Mantel,
Tony Shardlow, Ricardo Oliva, Marina Osipchuk, Kurt Lust.
Visitors:
David Chillingworth, Laurette Tuckerman, Gabriela Gomes, William
Langford, Steve Strogatz, Fernanda Botelho.
I wrote a set of notes based on the these discussions; these
notes can be accessed from my web site.
The following lists a few of my other activities during my first
year at the IMA.

I gave a talk in the Math Department's "Dynamics and
Mechanics" seminar organized by Rick Moeckel. Title:
Stability of the relative equilibria in a class of rotationally
symmetric mechanical systems.

I prepared a report based on the Industrial Seminar presented
by Nicholas Tufillaro of HewlettPackard on "Symbolic
Dynamics in Mathematics, Physics and Engineering." This
report is to become part of the IMA web site.

I attended the conference "Modeling and Analysis in Medicine
and Biology" at the University of Michigan in Ann Arbor.

I refereed papers for Physica D and the American
Journal of Physics.
Annual Program Organizers
Workshop Organizers
Visitors Postdocs
