**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**

UMD Collaborators: **Brian Hunt, Mike
Roberts **(Ph.D.
UMD 2003)**, Guillaume Marcais, Suzanne **

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

In collaboration with **Brandy
Rapatski **(Ph.D. UMD 2004) and

HIV entered the

**Chaos Projects**

UMD Collaborators:** Chris Danforth **(Ph.D. 2006)

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)

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

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, 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.