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## Research Accomplishments in Mathematics in 1997-98: Emerging Applications of Dynamical Systems

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 1997-98 program year on "Emerging Applications of Dynamical Systems." He also co-organized 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 11-15, 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 well-known 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 mini-review 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 1997-98 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 1997-98 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 1998-99 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 one-on-one 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 1997-98, a continuation of a long-standing problem in connecting the science and engineering disciplines with the biological community. However, during the 1998-99 year on Mathematicas and Biology, striking advances were made. More than 50 reseachers in medicine and biology at the University took part in the 1998-99 program. An ongoing Mathematics-Physiology 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, co-sponsored 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 math-biology links. Additional joint math-biology 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 Large-Scale 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 Large-Scale 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 1997--1998 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 15-19, 1997. The organizing committee of this workshop consisted of Eusebius Doedel (chair), Wolf-Juergen Beyn, Bernold Fiedler, Yannis Kevrekidis, and Jens Lorenz. The workshop concentrated on complex computational issues in dynamical systems. While computational techniques for low-codimension local bifurcations in few-degree 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, self--organized 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 29-October 3, 1997. It was organized by Laurette Tuckerman (chair), Edriss Titi, Herbert Keller, and Don Aronson. The numerical study of low-dimensional 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 large-scale physical problems as dynamical systems which, although high-dimensional, undergo low-codimension bifurcations. Applications of special interest include selected problems arising in fluid flows, and pattern formation in reaction-diffusion 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 Texas-Austin is the main organizer of the workshop on "Dynamics of Algorithms" held on November 17-21, 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 California-Santa Barbara and Jens Lorenz of the Department Mathematics and Statistics, University of New Mexico serve as co-editors 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 Large-Scale Dynamical Systems held on September 29 - October 3, 1997. Her report follows:

I. Open shear-driven flows

A. Perturbed plane Couette flow

I collaborated with Dwight Barkley (year-long IMA visitor for 1997-8) 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 three-dimensional (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 time-dependence and turbulence.

B. Symmetries in cylindrical wake flow

Dwight Barkley (year-long visitor) and Ron Henderson (participant in IMA workshop on Large-Scale Dynamical Systems in September-October 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, co-organizer 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, 1997-8) 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 spatio-temporal, 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 spatio-temporal 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 September-October 1997) and Alain Bergeon (participant in IMA workshop on Large-Scale Dynamical Systems), I completed a survey article on surface-tension-driven (Marangoni) convection due to combined thermal and concentration gradients. Henry, Bergeon, and I also continued work on Marangoni convection in three-dimensional 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 (year-long IMA visitor during 1997-8).

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 codimension-two point, at which a curve of Hopf bifurcations was annihilated by joining a curve of steady bifurcations, accompanied by the existence of heteroclinic infinite-period 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 off-diagonal 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 saddle-node 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 year-long 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 time-steppers, in Numerical Methods for Bifurcation Problems and Large-Scale 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 double-diffusive convection with opposing horizontal thermal and solutal gradients Phys. Fluids. 10, 850-858 (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, Three-dimensional 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 long-term 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 three-dimensional excitable media. Two examples of such media are the Belousov-Zhabotinsky 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 two-dimensional excitable media. Essentially nothing was known, however, about the effect of parametric forcing in three-dimensional 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 so-called axisymmetric scroll ring and the twisted scroll ring. We determined to high accuracy the spatio-temporal dynamics of both structures with and without forcing. Most significantly we were able to understand the dynamics through normal form equations based on non-compact 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, so-called 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 large-scale bifurcation methods based in time-stepping algorithms [3].

IMA visitors M.G.M. Gomes, R. Henderson, and I studied three-dimensional transition in the separated flow generated by a sudden expansion in an otherwise parallel flow channel. This geometry is known as a backward-facing step. We found that the first instability encountered as the Reynolds number is increased leads directly to a three-dimensional state, and that surprisingly the flow remains linearly stable to two-dimensional disturbances for very large Reynolds number [4].

I developed a heuristic description of nonlinear three-dimensional 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 large-scale 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 scroll-wave patterns in three-dimensional 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, "Three-dimensional instability in flow over a backward-facing step" in preparation for J. Fluid Mech.

[5] D. Barkley, "Nonlinear stability theory for three-dimensional 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 TEX 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 24-25, 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 non-compact 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 three-dimensional, time-dependent 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 blow-up 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 Hopf-type 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 blow-up and slow-fast 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 three-dimensional, non-planar 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 Belousov-Zhabotinsky 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 CO-oxidation 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 higher-dimensional 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 not-so-planned seminar in a wonderful working environment.

Wulfram Gerstner of Swiss Federal Institute of Technology Lausanne, Centre for Neuro-mimetic Systems was a participant in the workshop: Computational Neuroscience held on January 14-23, 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 long-term 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: Three-dimensional instability in flow over a backward-facing step (with Dwight Barkley and Ronald Henderson) We performed a three-dimensional 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 two-dimensional flow is a steady three-dimensional 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 15-20 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: Black-eye 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 two-dimensional. However, beyond onset a structure that does not typically occur in two-dimensional domains was observed - the black-eye pattern. In this letter we use the full three-dimensionality of the patterned layer to find a setting where black-eye patterns naturally occur. We propose that black-eye 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 symmetry-breaking 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 (cross-immunity) 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 co-circulate? We will try to answer some of these questions by modeling the mechanism of strain formation as a symmetry-breaking 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 Marangoni-Benard 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 co-organizers 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 Kuramoto-Sivashinsky 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 two-dimensional 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 Kuramoto-Sivashinsky Equation

Up to now estimates for this dimension have typically been of the form dim cLb, 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 Navier-Stokes 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 Kuramoto-Sivashinsky Equation (with M. Johnson and I. Kevrekidis)

We have used the visualization of two-dimensional 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, Jean-Pierre Eckmann, Henri Epstein, and Joachim Stubbe. A global attracting set for the Kuramoto-Sivashinsky equation. Comm. Math. Phys., 152(1):203-214, 1993.

[2] Ciprian Foias, Oscar P. Manley, and Roger Temam. Iterated approximate inertial manifolds for Navier-Stokes equations in 2-D. J. Math. Anal. Appl., 178(2):567-583, 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 241-277. Kluwer Acad. Publ., Dordrecht, 1993.

[4] Klaus Kirchgassner. Wave-solutions of reversible systems and applications. J. Differential Equations, 45(1):113-127, 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):631-642, 1993.

[6] R. Rosa. Approximate inertial manifolds of exponential order. Discrete and Continuous Dynamical Systems, 1:421-448, 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(3-4):1095-1108, 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:

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 finite-dimensional 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 IMA-related 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 1995-96 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 so-called "multi-cellular 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 non-radial 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 10-day 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 real-world 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, well-defined academic exercises found in classrooms, but rather they consisted of stimulating open-ended 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 signal-launching onto multi-mode 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 non-necrotic 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 coagulation-fragmentation 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 22-31, 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 14-23) 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, TU-Muenchen 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 14-23, 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 year-long 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 long-term 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 e-mail) 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 three-dimensional 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.

Small-world networks:

In early June, much of my time was spent dealing with the media excitement about my Nature paper with Duncan Watts on small-world 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 (co-organizer), 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 solid-state 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 'small-world' networks," Nature 393, pp.440-442 (1998).

[4] Steven H. Strogatz, "Nonlinear dynamics: Death by delay," Nature 394, pp. 316-317 (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 CNR-Universita 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 temperature-dependent strain energy function for a crystal exhibiting tetragonal-orthorhombic-monoclinic ('t-o-m') martensitic transformations. A very well-known material undergoing these phase transitions is for instance zirconia (ZrO2), which is the main toughening agent in transformation-toughened 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 t-o-m triple point in its pressure-temperature 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' Landau-type energy for the t-o-m crystals, as the lowest-order 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 product-phase variants) coexist. Our energy function is suggested in part by the analysis of a simple discrete mechanical system with four particles connected by six Lennard-Jones springs, which shows instabilities and bifurcations analogous to those characteristic of the t-o-m 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 t-o-m 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 symmetry-breaking transformations involving solid phases with a progressive reduction of symmetry.

There are several mid- and long-term 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 pressure-temperature 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 pressure-temperature 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 1997-98 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 well-organized 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.

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

2. Completed my article on the Entropy Formula for SRB-measures 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.

3. 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.

4. 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.

5. Initiated a joint work with Rafael de la Llave on the smooth dependence of SRB-measures 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 Finite-Dimensional Approximation Commun. Math. Phys. 193, 675-711 (also IMA Preprint series 1525)

[2] M. Jiang 1999 The entropy formula for SRB-measures of lattice dynamical systems, to appear, J. Stat. Phys.

[3] M. Jiang, Quasi-Monte Carlo studies of spatial averages of quadratic maps, 1998 1-7 + 16 Illust., IMA Preprint series 1584, submitted.

[4] M.Jiang, Dynamics and bifurcations of a 3D system modeling thermal instability, 1998 1-13 + 6 Illust., IMA Preprint series 1583, submitted.

[5] M. Jiang and R. de la LLave, Smooth Dependence of Thermodynamic Limits of SRB-measures, 1-34, 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 one-year position as a postdoc at the IMA's year on Emerging Application of Dynamical Systems, I had the privilege to interact with an excellent cross-section 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 real-time demonstration of SCIGMA, a software application which was developed in collaboration with Michael S. Jolly (one of the program's year-long 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 three-dimensional approximate inertial form for the one-dimensional periodic Kuramoto-Sivashinky equation. These observations were made by computing and observing the interaction of collections of two-dimensional 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 Kuramoto-Sivashinsky equation". When this work was still in progress, I delivered a Postdoc Seminar in April, 1998, called "Interacting two-dimensional manifolds and global bifurcations for the Kuramoto-Sivashinsky Equation."

3. The long-time dynamics of the Kuramoto-Sivashinsky 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 long-time 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 1997-98 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 low-dimensional 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 large-scale 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 (1991-1992) 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 large-scale problems. Instead of doing a bifurcation analysis of the large-scale problem, a low-dimensional 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 large-scale model was the Navier-Stokes 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 black-box modeling using artificial neural networks and a traditional Galerkin projection of the Navier-Stokes 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 large-scale 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.

Rolf-Martin Mantel is one of the 1997-98 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 post-doc 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 co-written with Dwight Barkley of the University of Warwick, who was also a year-long visitor at the IMA. Its title is "Parametric forcing of scroll-wave patterns in three-dimensional 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 1997-98 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 industry-related research and collaboration with the IMA participants for the program year. In the course of the project at the HTC we developed a low-cost 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 reaction-diffusion 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 "fiber-like" quality along with globally provided, space-based communication are the distinctive features of the broadband low-Earth 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 on-board 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 non-oriented 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 infinite-dimensional 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 finite-dimensional control.

A nonlinear model reduction method of the discretized PDE was employed to approximate the long-term 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 closed-loop 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 reaction-diffusion 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 differential-algebraic equations play in modeling chemical reactions and another focused on using reaction-diffusion 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 non-simple 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 f1(s) and f2(s), the perturbation from the initial shape in the horizontal direction will be denoted by ui(s) and the perturbation from the initial shape in the vertical direction will be denoted by wi(s). The coordinates of the beams after displacement will be given by xi=s+ui(s) and yi=fisub>i(s).

The first model which we considered allowed only vertical displacement of the beam (ui(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, s1 and s2, and forces it to a prespecified location y0 with a prespecified slope y0'. The clamping problem is modeled by two fourth-order ordinary differential equations subject to left end boundary conditions, continuity conditions at si, and the clamping conditions. This model problem has an analytic solution.

Unlike the clamping procedure, the welding procedure does not specify the position y0 or the orientation y0' 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 fourth-order 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 (ui(s) 0, wi(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 fourth-order 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 Cahn--Hilliard and Allan--Cahn 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 Cahn--Hilliard 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 Allan--Cahn equation forced by additive space--time white noise and the development of an SDE to account for the motion of the fronts. Previous work on the deterministic Allan--Cahn 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 Allan--Cahn 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 1997-98 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 1995-97. 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 high-quality 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 long-term 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 multiple-scale 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 1997-98. 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 long-term 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
"Row-switched states in 2-D underdamped Josephson junction arrays" Physical Review B, vol.57, no.17 (1998 May 1) 10893-10912.

[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) 767-768 (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.) 1-7.

[4] H. S. J. van der Zant and S. Watanabe,
"Dynamics of kinks and vortices in Josephson-junction 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), 283-302.

Warren Weckesser of Rensselaer Polytechnic Institute is one of the IMA Postdocs. He is on his 1st year of a two-year 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 non-identical Van~der~Pol-like 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 co-chair 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 Hewlett-Packard 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