**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

** **

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

HIV entered the

Chaos Projects

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.

A Mathematical theory of observation,

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

x_{n} = f(x_{n-j})
+ g(x_{n-1},...., x_{n-k})

where f(x) = a – x^{2} with a
> 2 and g is small, and 1 < j < k. Note that x_{n} = f(x_{n-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.