Current Research Projects

by J. Yorke

See also some reprints and preprints

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.

Period-Doubling Cascades Galore

with Evelyn Sander

 

Better methods for determining the genetic sequence of large genomes,

or how to assemble massive one-dimensional jig-saw puzzles

IPST genome assembly group

UMD Collaborators: Brian Hunt, Mike Roberts (Ph.D. UMD 2003), Guillaume Marcais, Suzanne Sind (Ph.D. UMD 2006), Poorani Subramanian, Aleksey Zimin (Ph.D. UMD 2003), James White

 

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 several assembly teams testing some new methods on the Drosophila genomes and we have recently completed an analysis of the rat genome.

 

Nonlinear dynamics of computer networks

            UMD Collaborators: Ian Frommer (Ph.D. UMD 2005), Brian Hunt, Ryan Lance (UMD 2005), Edward Ott, Amy F. Marinello, Russell Halper, Yiwei Chen

 

            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 (Ph.D. UMD 2004) and Frederick Suppe

 

HIV entered the U.S. population from African sources. It thus is surprising that the U.S. gay epidemic exploded over a decade earlier than the African one. We find that the same transmission dynamics explain the fast gay epidemic and the slow African one. The growth of HIV in a population is the combined effects of a fast wave of HIV transmission by newly infected persons and a slow wave of transmission by persons late in their illness. The propagation of HIV in a population is the combined effects of those fast and slow transmission waves.  The fast transmission wave was a significant factor in the spread of HIV among gays but not in the South-African epidemic. The difference in the two epidemics can be explained by ten-fold smaller effective contact rates (infectivity times frequency of contact) in the African epidemic compared to the gay epidemic.  We estimate how the infectiousness of a person varies as the disease progresses.  Our conclusions published in JAIDS in 2005 are important for the US population. Our conclusions mean health officials have time to intercept and get under treatment people who are infected, before they enter the “symptomatic” stage which is when they are most likely to infect people. In September 2006, the U.S. Centers for Disease Control announced a radical change in policy, recommending wide-spread screening of the U.S. population to detect as many HIV infected people as possible. In JAIDS September 2006 we attempt to explain why there are differing estimates of when HIV is most infectious. See also our discussion of this reversal on our web page HIV .

 

Chaos Projects

 

Chaos and weather prediction,

 

UMD Collaborators: Chris Danforth (Ph.D. 2006), John Harlim (Ph.D. 2006), Brian Hunt, Eugenia Kalnay, Eric Kostelich (Prof. at Arizona State U. visiting UMD), Ed Ott, Istvan Szunyogh, Aleksey Zimin (Ph.D. UMD 2004)

            All prediction is extrapolation. Our primary goal is to develop algorithms for finding the global initial conditions for weather prediction.  We are using nonlinear dynamics (or chaos) theory to develop better weather initialization 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. We run it at an intermediate resolution of about 3 million variables.

 

A Mathematical theory of observation,

 

            In collaboration with Will Ott (Ph.D. UMD 2004), Misha Brin.

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, Univ. of Delaware

 

            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, both of George Mason University

           

            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

            In collaboration with Helena Nusse and Joe Skufca (Ph.D. 2005)

 

            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. Work with Joe Skufca includes an investigation of the chaotic saddle of a 9 dimensional differential equation system representing Couette flow. It is in collaboration with Bruno Eckhardt.

 

A physical realization of the Plykin attractor

(or the dynamics of a taffy-pulling machine)

            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.  The mathematical question of why this process works and the candy does not develop week spots can be formulated as a nonlinear Frobenius-Perron operator, where the amount of stretch is greater when the density is lower.