Current Research Projects
by J. Yorke
This page may not be up to date and the projects are listed in no particular order!
Names of graduate student collaborators are in italics. (Mike Oczkowski, Mike Roberts, and Aleksey Zimin received Ph.D.s in 2003 and Will Ott in 2004)
Better methods for determining the genetic sequence of large genomes,
or how to assemble massive one-dimensional jig-saw puzzles
Collaborators: Kate Meloney, Suzanne Sindi, Cevat Ustun, Brian Hunt, Wayne Hayes, Mike Roberts, and Aleksey Zimin
The “shotgun method” is one of the ways to determine a genomic sequence, which consists of a string of bases, each represented by one of four letters: A, C, G, or T. Shotgun sequencing begin with the creation millions of small overlapping pieces of DNA called “reads”, each about 500 letters long. They are created without any information about where in the genome they came from, and as they are created and read, about 1% of the letters are reported incorrectly. The problem is to put these together into a nearly correct genome. It is a grand jig-saw puzzle. We have a joint effort with Celera Genomics testing some new methods on the Drosophila (fly) genome.
Nonlinear dynamics of computer networks
in collaboration with Ian Frommer, Ryan Lance, Brian Hunt, and Ed Ott
Transmission Control Protocol (TCP) is the mechanism responsible for controlling the rate of internet connections. There are many mathematical models which describe how this rate changes, some models are stochastic, others are deterministic. We are developing and evaluating deterministic models in terms of their ability to predict network traffic over short time spans. Our models are aimed at capturing the dynamics of an Internet connection which experiences congestion. We are comparing our results to those of other models, focusing on cases of irregular network behavior. An important step in evaluating such models is to analyze actual network traffic data. This is not only to understand what behaviors are present, but also to facilitate new empirical models. Analyzing this type of data is difficult because certain key rate-controlling variables are not directly recorded and must be inferred. We have novel methods of reconstructing such variables leading to a greater understanding of TCP dynamics and perhaps new types of models.
Modelling the population dynamics of HIV, or “Why the US Gay HIV Epidemic Exploded Years Before the Sub-Saharan Epidemic”
in collaboration with Brandy Rapatski and Frederick Suppe
HIV entered the
in collaboration with John Harlim, Brian Hunt, Eugenia Kalnay, Ed Ott, Mike Oczkowski, DJ Patil, Istvan Szunyogh, Aleksey Zimin
We are using nonlinear dynamics (or chaos) theory to develop better weather prediction algorithms for use with high performance computing. The project is based on the idea that the weather – at least as exhibited by weather models – is not terribly chaotic. We develop techniques for understanding existing, whole-earth weather models using ensembles of solutions, collections of solutions with slightly different initial conditions. We recently received a Keck Foundation grant to begin this project. Our favorite model, the one we investigate most intensively, was developed by the National Weather Service. It has about 3 million variables.
in collaboration with Will Ott.
When a laboratory experiment (like a moving fluid) is oscillating chaotically, the state of the experiment is revealed only by simultaneously measuring a limited number m of variables in the experiment, such as fluid flow rates at different points, or temperatures or other physical measurements. So-called “embedding” techniques have been developed where in the chaotic attractor can apparently sometimes be reconstructed. Is the number of variables m large enough to reconstruct the dynamics? Our goal is to justify these embedding methods, or rather to what is necessary for them to work. Ruelle and Takens introduced the notion of measuring the dimension of a chaotic attractor using such ideas. Is the dimension that we compute representative of the actual dimension of the attractor? Are Lyapunov exponents that are computed from data real; if several are computed, which are real and which are numerical artifacts? In The Republic, Plato has Socrates discussing the very limited nature of observation. He says we do not see reality but only limited images or shadows of reality. We must use these shadows to understand reality.
Topological Horseshoes and other topological phenomena
in collaboration with Judy Kennedy
Dynamical systems exhibit a wide variety of phenomena that must be studied topologically. When we studied topology behind Smale’s horseshoes, we found the idea was much more general than we had suspected and we founds many intriguing examples. Currently we are investigating some rather difficult topology in horseshoes for the difference equation
xn = f(xn-j) + g(xn-1,...., xn-k)
where f(x) = a – x2 with a > 2 and g is small, and 1 < j < k. Note that xn = f(xn-1) exhibits transient chaos and almost all trajectories diverge. The small perturbation g makes the problem depend on j-dimensional horseshoes.
Explosions of chaotic sets as a parameter is varied
in collaboration with Kathy Alligood and Evelyn Sander.
We have investigated with Carl Robert how chaotic sets can suddenly change, that is, explode, as a parameter of the system is varied. That explosions occur is an old concept. We believe there are a small number of situations that lead to explosions and that these can be characterized. We have largely done so -- for two-dimensional maps. Now we are investigating maps in one dimension and maps in dimensions higher than 2. It is a severe challenge to try to imagine what the intricate behaviors of these systems can be.
Developing tools for the numerical exploration of nonlinear dynamical systems
A long-time on-going collaboration with Helena Nusse
Our book “Dynamics: Numerical Explorations” includes a program that allows the user to carry our many kinds of investigations of dynamical systems, but developing new numerical techniques is an ongoing effort. Most scientists only became aware of chaos when they could visualize it with their computers. But any picture that is created with a computer might be thought of as a conjecture because we may not be certain what we are seeing, and we may wonder how much of the picture is numerical artifact. Examining pictures of dynamical systems is a constant source of inspiration and wonder. One of our techniques for basins of attraction has recently given a characterization of the basins with the most entangled boundaries. These are easy to characterize in computational terms! Computation often leads to surprises and new understanding.
in collaboration with J.T. Halbert
Early investigators of diffeomorphisms naturally focused on the simplest case first: uniformly hyperbolic maps. R.V. Plykin demonstrated in 1974 that nontrivial hyperbolic attracting sets exist for some of these maps. Does the type of attractor he found (called a Plykin attractor) arise in connection with a physical system? We hope to show that the answer is yes. We are currently studying the dynamics of a taffy-pulling machine. We expect to show that the action of this machine on taffy leads naturally to a diffeomorphism of an open set in the plane that has a Plykin attractor, if we are permitted some artistic license. It can also be reduced to an intriguing map on an interval, if we are granted a bit more artistic license. This work might have application in materials processing as when fibers like carbon nanotubes must be aligned with each other.