On systems of rational difference equations and periodic tetrachotomies
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.
Keywords
Cite
@article{arxiv.0909.4308,
title = {On systems of rational difference equations and periodic tetrachotomies},
author = {Frank J. Palladino},
journal= {arXiv preprint arXiv:0909.4308},
year = {2011}
}
Comments
The earlier work required that the matrix be Hermitian and so did not give the full characterization of qualitative behavior. This version improves on the work posted prior and gives the complete picture in this case