This paper is the first in a series of papers which will address, on a case by case basis, the special cases of the following rational system in the plane, labeled system #11. $$x_{ n+1}=\frac{ \alpha_{ 1}}{ A_{ 1}+y_{ n}},\quad y_{ n+1}=\frac{ \alpha_{ 2}+\beta_{ 2}x_{ n}+\gamma_{ 2}y_{ n}}{ A_{ 2}+B_{ 2}x_{ n}+C_{ 2}y_{ n}},\quad n=0,1,2,...,$$ with $\alpha_{ 1},A_{ 1}>0$ and $\alpha_{ 2}, \beta_{ 2}, \gamma_{ 2}, A_{ 2}, B_{ 2}, C_{ 2}\geq 0$ and $\alpha_{ 2}+\beta_{ 2}+\gamma_{ 2}>0$ and $A_{ 2}+B_{ 2}+C_{ 2}>0$ and nonnegative initial conditions $x_{ 0}$ and $y_{ 0}$ so that the denominator is never zero. In this article we focus on the special cases which are reducible to the Riccati difference equation.
Categories: math.DS