John M. Franks - US grants

This is a grant under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences. It is for the purchase of special purpose equipment dedicated to the support of research in the mathematical sciences. In general, this equipment is required by several research projects, and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers, and with substantial cost-sharing from the institution submitting the proposal. This is an instance of university, industrial, and government cooperation in the support of basic research in the mathematical sciences. The equipment in this project will be used for research projects in analytical and algorithmic studies for nonlinear systems, symbolic computations and special functions, homological algebra computations, and dynamical systems.

The three principal investigators will apply various geometric, symbolic dynamic, and ergodic methods to solve problems concerning finite-dimensional flows and mappings. One investigator will concentrate on fixed-point theorems for surface homeomorphisms, the conjugacy problem for subshifts of finite type, and knots in three-dimensional flows. A second investigator will study the creation of strange attractors through bifurcation at the onset of chaos. The third investigator will use ergodic theory to study the geodesic flow for manifolds with nonpositive curvature. "Dynamical systems" is the mathematics of movement. Two problems of intense interest in this field are the topological nature of closed orbits and bifurcations at the onset of chaos. One investigator will continue his analysis of the knot types of closed orbits in three dimensions. He seeks to use the Jones polynomial which catergorizes knot types to better understand orbits of such systems as the Lorenz and Henon attractors. A second investigator will improve upon our understanding of Smale "horseshoes" created at the onset of chaos. For a system of differential equations describing for example water flow, the existence of these bent topological objects would indicate turbulent behavior.

This award supports the establishment of the "Midwest Dynamical Systems Program Group." Sixteen mathematicians specializing in dynamical systems and the study of chaos, from Northwestern University, the University of Minnesota, and the University of Cincinnati, will continue their group activities while partially supported by this five-year award. Participant support costs of the Midwest Dynamical Systems Conference, which has run biannually since 1970, will be partially offset. Support for two graduate students, a postdoc, and compatible computer equipment are also included in the award. The mathematicians involved have a long history of close interaction and this award will help strengthen their ties.

Students are participating in a newly developed Nonlinear Lattice Dynamic course, one of a set of experimental mathematics courses being developed at Northwestern and one which cannot be taught without significant computer facilities. Displaying features of systems of equations on the screen in the classroom laboratory makes it possible to discuss and analyze how the whole family of solutions depend on their initial conditions and to compare and contrast how the nonlinear solutions are approximated by their linear counterparts.

9504760 Franks The proposed research pertains to various aspects of dynamical systems: the dynamics of surfaces maps, stability theory for various flows, applications to celestial mechanics and Hamiltonian systems are among the topics to be pursued. Dynamics of surfaces flows - stability issues as well as area conservation issues - is directly relevant to understanding many natural phenomena and predicting their long time behavior. For example, the proposed research may lead to a better understanding of large scale fluid flow problems arising in ocean dynamics involving shallow water waves.

Professor Franks will continue his investigation of the dynamics of surface maps, especially area preserving ones. Questions concerning the existence of periodic behavior in area preserving two-dimensional discrete dynamical systems will be addressed. Professor Xia's proposed research concentrates on the following three aspects of dynamical systems: (1) the dynamics of the Newtonian n-body problem and celestial mechanics; (2) hyperbolicity and bifurcations in global dynamics; and (3) Hamiltonian dynamics and symplectic diffeomorphisms. Surface maps are two-dimensional transformations; the study of such transformations, especially area preserving ones, has a long history going back to Poincare and G. D. Birkhoff. There are numerous applications of results in this area to classical mechanics as well as to more modern chaotic dynamics. In physical terms, Professor Xia's proposed research is on: the dynamics of bodies being acted on by gravitation; the way the dynamics of a system changes as parameters in it vary; and the dynamics of systems in which energy is conserved.

This is a proposal to investigate aspects of low dimensional dynamical
systems. The most important long term project is to better
understand a topological description of C^r generic area preserving
diffeomorphisms of surfaces where r > 1. In addition the proposer
hopes to investigate the action of more general groups than the integers
or the reals on surfaces, and in particular area preserving actions.

Professor Franks will continue his investigation of the
dynamics of surface maps. The study of such transformations,
especially area preserving ones, has a long history going back to
Poincare and G. D. Birkhoff. There are numerous applications of
results in this area to classical mechanics as well as more modern
chaotic dynamics. The proposed research will address questions
concerning the existence of periodic behavior in area preserving
two-dimensional discrete dynamical systems and a topological
description of the dynamics of such systems.

Abstract for proposal DMS-0555463:


This project investigates aspects of low dimensional dynamical
systems. The study of such transformations, especially area
preserving ones, has a long history going back to Poincare and
G. D. Birkhoff. In particular the project considers smooth group
actions on surfaces and the relation between the algebraic properties
of the group and the dynamics the action exhibits. A related question
addressed by this proposal is the question of the existence of global
fixed points for two-dimensional dynamical systems and how this
relates to the algebraic properties of the system.


This project investigates aspects of dynamical systems on
surfaces. There are numerous applications of results in this area to
broader fields of science, especially to classical mechanics and more
modern chaos theory. The novelty of the proposed research is that it
addresses the relationship between the algebraic nature of dynamical
systems called ``group actions'' and the geometric or topological
behavior they exhibit. Anticipated results from this proposal will
advance our knowledge of dynamical systems and will explore new
relationships between dynamics and algebra.

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).

This is a project to investigate aspects of low-dimensional dynamical systems. The proposed research addresses the connection between algebraic properties of a group that acts on a surface in an area-preserving way and the possible topological nature of the dynamics of the action. One theme is to try to find global fixed points for a smooth group action and use the induced representation of the group into the automorphisms of the tangent space at the fixed point to conclude information about the action. Anticipated results from this project will advance our knowledge of dynamical systems and will explore new relationships between dynamics and algebra.

This proposal concerns transformations of surfaces as they evolve in time. The study of such transformations, especially area-preserving ones, has a long history going back to work of Henri Poincare and G. D. Birkhoff, work that was motivated by problems in celestial mechanics. For such a system a "state" is a point on a surface and the objective is to understand how the collection of all states evolves in time. Time can be considered as either continuous (represented by a real number) or discrete (represented by an integer). The present project deals with the discrete case and generalizes it to consider evolutions where the analogue of time is represented, for example, by a matrix rather than an integer. This is part of a long-term program to understand the relationship between the algebraic nature of dynamical systems and the geometric or topological behavior they exhibit. There are numerous applications of results in this area to broader fields of science, especially to classical mechanics, celestial mechanics, and more modern chaos theory.

This award will support activities during a planned "Emphasis Year in Dynamical Systems" at Northwestern University during the 2010-11 academic year. The emphasis year, funded in part by the Northwestern Department of Mathematics Department, will feature both long- and short-term visiting scholars, postdoctoral positions in dynamical systems, and the annual Midwest Dynamical Systems Conference.

The semester's programs are specifically organized around the research areas of ergodic theory, discrete group actions, Hamiltonian dynamics, and smooth and geometric dynamics, with an organizing theme of "Trends in Dynamics for the Coming Decade." This theme will guide the events of the year and will be the title and focus of a concluding major conference to be held in Spring 2011. The funding will be used to support both visitors to the department and participants in the conference, increasing collaboration within and across disciplines. Much of the funding will be directed to young mathematicians, particularly postdoctoral fellows, graduate students, and junior faculty, who do not have their own sources of support.