The Dynamical Systems Group at Boston University

by Tasso Kaper, Boston University

The dynamics group at Boston University was founded in the late 1970s. At its core, the dynamics group consists of seven faculty, approximately 15 PhD students, and a number of postdocs. More broadly, participants in dynamics group activities also include many of the postdocs and students of the Center for BioDynamics -- which has as its primary mission the advancement of applications of dynamical systems in the biological and engineering sciences.

This photograph was taken Monday April 11, 2005 outside the Mathematics Department of Boston University.
Front row (from left): Kalin Kostadinov, Sebastian Marotta, Margaret Beck, Dick Hall, Elizabeth Russell, Dan Look, and Mamikon Ginovyan.
Second row (from left): Horacio Rotstein, Bob Devaney, Marina Bevzushenko, Eric Wahl, Oleg Mikitchenko, Elizabeth Zollinger, David Uminsky, Cecilia Behn, and Chelsea Smock.
Third row (from left): Ehud Sivan, Paul Blanchard, Nancy Kopell, David Fried, Gene Wayne, Steve Epstein, and Jozsi Jalics.
Back row (from left): Matt Holzer, Gabriel Soto, Nikola Popovic, Tasso Kaper, Baldur Hedinsson, Russell Jackson, and Peter Barendse.

The group's research and training missions span topics in pure and applied dynamical systems, ranging from complex dynamics, dynamical zeta functions, and Hamiltonian dynamics to rhythms in the nervous system, invariant manifolds for PDEs, and multiscale systems. Faculty members and their primary dynamics interests are:

Paul Blanchard:
	complex analytical dynamical systems, dynamics of Newton's method, rational maps whose Julia sets are Sierpinski curves, differential equations education.
Bob Devaney:
	complex analytic dynamical systems, iterations of rational and entire functions, chaotic dynamics, structures of Julia sets especially those that are Sierpinski curves, problems at the interface of topology and dynamics, singular perturbations of analytic maps.
David Fried:
	dynamical zeta functions, periodic orbits, hyperbolic flows, Reidemeister torsion, and symbolic dynamics for geodesic flows on noncompact hyperbolic manifolds.		The ODE project
Dick Hall:
	celestial mechanics, coorbital moons (Janus and Epimethius of Saturn), shepherding moons (e.g., in the F-ring of Saturn), triple collision problems, Hamiltonian systems, geodesic flows, twist maps.
Tasso Kaper:
	systems with multiple time and/or length scales, geometric singular perturbation theory, stability and bifurcation of patterns in activator-inhibitor systems, model reduction in chemistry, renormalization groups methods in perturbation theory, dynamics of gas bubbles.
Nancy Kopell:
	rhythms of electrical activity in the nervous system, their origins, coherence, and functional implications, applications to olfaction, auditory cortex, neocortex, schizophrenia, and the mechanisms of anesthetics, invariant manifold theory for multiple scale systems, canards in localization phenomena and in coupled oscillators, applications of geometric singular perturbation theory to a variety of problems.
Gene Wayne:
	conservative and dissipative PDEs, KAM theory for Hamiltonian PDEs, invariant manifolds of solutions of the Navier-Stokes equations, validity of the KdV and other approximations for interacting water waves, invariant manifolds for dissipative PDEs.

The group has had a dedicated postdoc position for the past 20 years -- which has rotated among the subdisciplines -- as well as some NSF postdocs. A partial listing includes: Amitabha Bose (N.J.I.T.), Arek Goetz (San Francisco State U.), Russell Jackson (USNA), Kresimir Josic (U. Houston), and Bruce Peckham (U. Minnesota, Duluth). Also, former CBD postdocs are listed on the CBD website.

Postdoc Guillaume van Baalen recently completed his thesis with Jean-Pierre Eckmann (Geneva) and works on invariant manifolds for PDEs, especially for the Navier-Stokes equations, on phase turbulence in the complex Ginzburg-Landau equation, and on the Kuramoto-Sivashinsky equation. Postdoc Nikola Popovic recently completed his thesis with Peter Szmolyan (Vienna) and works on singularly perturbed systems, geometric desingularization, rigorous asymptotics, stability of weak shocks, problems with logarithmic switchback terms, a gene regulatory network problem, and on micromagnetics.

Besides the research and guiding of postdocs, a central group mission is the mentoring of dynamics students. This is done both in individual research groups as well as in jointly-sponsored groups, with many students working with more than one advisor. Also, the students are actively engaged with teaching each other, mainly through their weekly student seminar, which they have been running for the past several years. These activities -- along with social activities -- have greatly helped to promote group cohesion.

The group sponsors a weekly dynamics seminar on Mondays. This features a mix of internal, regional, national, and international speakers, as well as alumni. Of course, the real reason to host these seminars is for the faculty to learn more about each other's topic areas. For example, Bob Devaney still wonders why all reaction-diffusion equations involve cubic nonlinearities.

Nonlinear dynamics also flourishes in other departments at BU. Provost and physics professor David Campbell leads a research group in physics and continues to serve as editor-in-chief of the AIP journal Chaos. Physics professor Sid Redner was the Ulam Scholar at the Center for Nonlinear Studies at Los Alamos National Laboratory in 2004-5 and is active in the CBD. Biomedical Engineering professor Jim Collins co-directs the CBD and has a large research group working on various applications of dynamical systems and mathematics to problems in gene regulatory networks, noise-based sensory prosthetics, and human balance control. He has also carried out mathematical work on stochastic resonance, symmetry in gaits, and attractor reconstruction from short experimental time series.

Finally, several group members are active in educational efforts involving dynamics and differential equations. For example, Bob, Dick, and Paul direct the ODE project, which promotes the use of ideas from the modern theory of dynamical systems in the traditional sophomore-level course in differential equations. Their textbook stresses qualitative, analytic, and numerical approaches and will soon come out in its third edition.

		Related educational endeavors include the Dynamical Systems and Technology Project, led by Bob, which is designed to help secondary school and college teachers of mathematics bring contemporary topics in mathematics into the classroom, and to show them how to use technology effectively in this process. The tip of the iceberg is visible at http://math.bu.edu/DYSYS/ where a series of JAVA applets are up and running that allow the user to explore chaos and fractals. Also, Bob runs two Math Field Days every Fall for high school students, and these often feature Dick and his box of symmetry toys.
Dancing Triangles

All in all, we strive to maintain a collegial group within a collegial department. In lighter moments, we may think of ourselves as being the analogs of some central dynamics phenomena: Bob and his group clearly form a chaotic attractor, since a series of students working on theses with other dynamics faculty have worked with him on some aspect of complex dynamics. David recurrently discovers the underlying geometric and algebraic structures in the dynamical systems he studies. Dick is affectionately known as a mild-mannered, yet truly celestial mechanician by his students. Gene can find a smooth manifold for just about anything and always works in the right spaces, or maybe it's better to say that he works on complicated equations in easy spaces, which makes him the envy of the geometers who work on simple equations in complicated spaces. Nancy has become the wizard of oscillations, and her interests transversely intersect just about everyone's in the group. Paul gently supplies the institutional memory function, and Tasso is normally a hyperbolic guy who finds it useful to spend time near slow manifolds to get research done.