On rational systems in the plane. I. Riccati Cases

TL;DR

This paper investigates specific rational systems in the plane that can be reduced to Riccati difference equations, providing a detailed case-by-case analysis of their dynamics and solutions.

Contribution

It offers the first detailed analysis of particular rational planar systems reducible to Riccati equations, expanding understanding of their behavior and solution structure.

Findings

Identification of cases reducible to Riccati equations

Analysis of stability and dynamics in these cases

Explicit solutions for certain parameter configurations

Abstract

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.