On systems of rational difference equations and periodic tetrachotomies

TL;DR

This paper analyzes a system of rational difference equations with nonnegative parameters, establishing conditions for periodic tetrachotomy behavior based on a specific 2x2 matrix, expanding understanding of complex dynamic behaviors in such systems.

Contribution

It introduces a new class of rational difference systems with specific zero-structure and characterizes their long-term periodic behaviors through matrix analysis.

Findings

Existence of periodic tetrachotomy depending on matrix properties.

Conditions for stability, periodicity, or divergence of solutions.

Framework for analyzing similar rational difference systems.

Abstract

We study the following system of two rational difference equations x_n=({\beta}_k x_(n-k)+{\gamma}_k y_(n-k))/(A+\Sigma_(j=1)^l[B_j x_(n-j) ]+\Sigma_(j=1)^l[C_j y_(n-j) ]), n \in N, y_n=({\delta}_k x_(n-k)+\in_k y_(n-k))/(q+\Sigma_(j=1)^l[D_j x_(n-j) ]+\Sigma_(j=1)^l[E_j y_(n-j) ]), n\in N, with nonnegative parameters and nonnegative initial conditions. We assume that B_j=C_j=D_j=E_j=0 for j=k, 2k, 3k, ...and establish the existence of periodic tetrachotomy behavior which depends on a 2X2 matrix with entries {\beta}_k, {\gamma}_k, {\delta}_k, and \in_k.