Website for Global Asymptotic Stability paper

by Emilie Hogan and Doron Zeilberger


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.



Web Books:

Computer generated proofs:

    For each order 1 rational difference equation with parameters ( e.g. xn+1=1/(A+xn) ) 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`;