# Website
for Global Asymptotic Stability paper

### by Emilie Hogan and Doron Zeilberger

### Abstract:

Global asymptotic stability of rational difference equations is an area
of
research that has been well studied. In contrast to the many current
methods
for proving global asymptotic stability, we propose an algorithmic
approach.
The algorithm we summarize here employs the idea of contractions. Given
a
particular rational difference equation, defined by a function $Q$
which maps
the $k+1$ dimensional real numbers to itself, we attempt to find an
integer,
$K$, for which $Q^K$ shrinks distances to the difference equation's
equilibrium
point. We state some general results that our algorithm has been able
to prove,
and also mention the implementation of our algorithm using Maple.

### Paper:

GAS_Hogan_Zeilberger.pdf

### Web Books:

#### Computer generated proofs:

For each order 1 rational difference
equation with parameters ( e.g. x_{n+1}=1/(A+x_{n})
) 50 sets of
parameter values were chosen. For each of the parameters the Maple
program first verifies that the difference equation is Locally
Asymptotically Stable. If it is not then the program stops. If it is
then the program incrementally checks K values starting with 1 and
ending with 9 in order to prove Global Asymptotic Stability. If one of
the K values works then the proof of Global Asymptotic Stability is
given.

The (large) pdf is found here
(604 pages, 1.69 MB).
#### Computer generated results:

Similar to the above, but with proofs omitted. If a
K value is found then we don't show the proof. The proof can be
produced, but the documents would be much too large.

Here is the file for Order 1
rational difference
equations.

Here it is for Order 2
rational difference
equations. Note that two difference equations are missing from this
file at the moment.

### Maple Code:

The maple code used to produce the above Web Books
and to investigate the topic in general can be found here. To use it, download the
.txt file, open Maple, and
type

read
`<directory>\\ProveGAS.txt`;

where <directory> is where you have saved the file. Notice
that in the directory single slashes "\" must be replaced by double
slashes "\\". For example, if you save the file to the directory
C:\maple\GAS you will type:

read
`C:\\maple\\GAS\\ProveGAS.txt`;