IJPAM: Volume 59, No. 4 (2010)

ON THE GEOMETRY OF A FOUR PARAMETER MIXED
RATIONAL/LINEAR SYSTEM OF PLANAR
DIFFERENCE EQUATIONS

Aaron S. Clark
Department of Mathematics and Statistics
University of Nebraska at Kearney
Founders Hall, Kearney, NE 68847, USA
e-mail: clarka1@unk.edu


Abstract.Motivated by questions posed by Ladas et al, we provide a concise geometric analysis of the rational, planar system of difference equations

\begin{eqnarray*}
x_{n+1} &=& \displaystyle\frac{a + b y_n}{x_n} ,\\
y_{n+1} &=& c x_n + d y_n,
\end{eqnarray*}

with initial values $x_0 > 0$ and $y_0 \geq 0$ and non-negative real parameters $a$, $b$, $c$, and $d$. This is accomplished by associating a planar mapping $F$ to the system in such a way that the orbit structure of $F$ gives information about solutions to the system. Consequently, we are able to classify the behavior of the system across its 4-dimensional parameter space for all possible initial data.

Received: February 2, 2010

AMS Subject Classification: 39A10, 65Q10

Key Words and Phrases: difference equation, rational difference equation, planar system

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 59
Issue: 4