Research Statement for Dr. Jeffrey Feuer
Visiting Lecturer at St. Michael’s College
My research deals primarily with discrete dynamical systems, also known as difference equations.
I have been involved with this field since 1994, my second year of graduate school (post masters) at the
University of Rhode Island. I got involved with difference equations because of my thesis advisor, Dr.
Gerry Ladas. His enthusiasm for research is infectious and I hope I can one day emulate him. To a small
extent, I will get to try to emulate him next semester, when I will run a one credit senior seminar in
mathematics.
My first project in difference equations began when Dr. Ladas posed an open problem during our
research class. The problem was to find an invariant for a non-autonomous version of a difference equation
known as Lyness’ Equation (which arose as a number theory problem in 1942):
xn  1 
xn  A
,
xn  1
where n is an integer greater than or equal to -1, A is a positive real parameter and the initial conditions (x-1
and x0) are arbitrary positive real numbers. An invariant, analogous to a first integral for a differential
equation, is used to prove solutions are bounded (do not tend towards infinity) and persist (do not get
arbitrarily close to zero). It is a function of two variables (x n-1 is like x, xn is like y) for a second order (xn+1
does not count towards the order) difference equation like Lyness’ Equation. No matter which two
consecutive terms in the solution are plugged in for the two variables, you get the same value for the
invariant function. For example, if the solution was {...2,3,4,5,...}, and I(x,y) was the invariant function,
I(2,3)=I(3,4)=I(4,5)=c, where c is a constant. The consecutive terms of any specific solution all lie on one
level curve of an invariant function. An autonomous difference equation is one where the parameter(s) do
not depend on the term in the solution. To make Lyness’ equation be non-autonomous, one would replace
the A with An.
In order to develop the invariant for the non-autonomous version of Lyness’ Equation where An is
periodic with period 3, I looked at the invariant for the autonomous version of the equation, as well as the
one for the non-autonomous version where An is periodic with period 2. I was able to generalize the
invariant and in the process developed a whole new technique for finding invariants! Every invariant we
have ever seen, regardless of the difference equation, can be derived using this method. I used this
technique to find invariants for non-autonomous versions of the Lyness’ Equation as well as for the nonautonomous versions of several generalizations. The most common generalization we studied was:
xn  1 
Axn  B
,
(Cxn  D) xn  1
where n is an integer greater than or equal to -1; A,B,C, and D are positive real parameters and the initial
conditions are arbitrary real numbers. This equation is sometimes referred to as the “Generalized Lyness
Equation”. As part of my thesis, I generalized Lyness’ Equation and the Generalized Lyness Equation to
any dimension. My method found all the invariants of these generalizations, regardless of order!
In another part of my thesis, I displayed invariant regions for Lyness’ Equation and the
generalization to any dimension. If we make a graph of the solution of a second order difference equation,
we plot each pair of consecutive terms in the solution as the x coordinate and y coordinate respectively (if
the solution was{...2,3,4,...}, we’d plot (2,3) and (3,4) as two of the points in the graph, called the phase
plane). In general, you make a phase plane graph in the same number of dimensions as the order of the
equation. A third order difference equation would be plotted with three consecutive terms in the solution in
a three-dimensional graph. An invariant region would be an area of the graph that if one point plotted in
the phase plane was inside the region, all the points plotted would be inside the region. One important
consequence of having an invariant region is that solutions are automatically bounded and they also persist
in this case (because it turns out that the exact regions do not include the origin or either axis). In the last
part of my thesis, I showed how the reader can use the invariants, which I had previously found, for the
non-autonomous version to find every possible invariant region for this equation. The technique is one that
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Research Statement for Dr. Jeffrey Feuer
Visiting Lecturer at St. Michael’s College
the reader may use for other difference equations, as long as he or she has an invariant.
Just before I graduated, Dr. Ladas and I started a new project with two graduate students. Dr.
Ladas had me supervise the work of these two graduate students, especially the one who was working on
his first project with us. This project, which led to my fourth published paper (all the citations for my
papers can be found in my vitae) deals with a difference equation sometimes referred to as the “period 7
max equation”:
xn  1 
max{ xn, A}
,
xnxn  1
where n is a positive integer greater than or equal to -1, A is a positive real parameter and the initial
conditions are positive real numbers. The name derives from the fact that when A=1, every solution is
periodic with period 7. There is a similar paper dealing with the “period 5 max equation”, more commonly
referred to as the “Lyness max equation”. For this equation, the solutions are periodic with period 5 when
A=1. We read the paper and quickly generalized most of the results to our equation. One of the results I
wanted to generalize was the discovery of many periodic solutions when A is not 1. I was able to quickly
come up with the results, but a general proof eluded me for about a year. I could have generalized the
results from the period 5 equation, but I knew that I could show more. I wanted to show that I had all the
possible periods and all the possible solutions which are periodic with a given period. While I was at
Merrimack College, in 1998-99, I was finally able to find a proof, and the idea can be used for other similar
equations. I was able to use this idea to show that the results in my colleagues’ paper on the Lyness max
equation were all the possible periods, and they had all the solutions that could be found. I have submitted
a paper for publication (my fifth) based on this work. I put the results for the period 7 equation at the end
of this paper.
The proofs for the period 5 max equation are fairly easily generalized to other equations. In fact,
earlier this year, some graduate students at the University of Rhode Island began work on the “period 8
max equation” (the xn in the denominator of the period 7 max equation is squared). I generalized my
results from the period 5 max equation and period 7 max equation to their equation and became a
collaborator on their project. They are actually involved in working with both the autonomous and nonautonomous versions of the period 8 max equation.
In order to help them with the non-autonomous version, I have started working on the nonautonomous version of the Lyness Max equation (the parameter A is now periodic) because the
autonomous version of the Lyness Max equation was far easier to work with than the period 8 max
equation. I have found that there are many more cases and so it has proven to be much more involved than
the autonomous equation, that I had previously studied. Given time I am confident I will finish the project!
When this project is over, I plan to work on some difference equations related to mathematical
biology. While attending the annual AMS/MAA meetings the last few years, I have had the occasion to
meet some of the “big names” in mathematical biology and more of the “big names” in difference
equations. I must say it was a great thrill getting to give talks during the same special sessions where they
have given talks. I have found many of the talks by the mathematical biologists very interesting. I have
read several papers by James Selgrade at the University of North Carolina. I have also read several papers
by John Franke also at the University of North Carolina, some of which were co-authored with Abdul-Aziz
Yakubu at Howard University. Dr. Yakubu actually posed some open problems at the joint meetings in
1999. Unfortunately, the projects that I was involved in this year did not leave me much time to make
much progress towards helping him. However, I have acquired reprints and preprints of some of his papers
and have spent some time reading through them. Basically, the problem that he and Dr. Franke have
worked on (together and separately) is studying predator-prey interactions. In the last few years, they have
been investigating the effects of a safe refuge on the populations, to see what happens if the prey has a
haven which some of them can use to elude the predators. I know that I have much to learn when I finish
my current project to catch up on the history of the model and exactly what progress has been made.
However, I find the material fascinating so I am looking forward to it!
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