Three Trichotomy Theorems

TL;DR

This paper introduces a unified periodic trichotomy theorem for a class of kth order rational difference equations, encompassing existing results and revealing new periodic behaviors.

Contribution

It develops a new unified trichotomy theorem that consolidates all known periodic behaviors of the equation into three main categories, including new examples.

Findings

Unified trichotomy theorem for the difference equation

Identification of three major periodic behavior families

New examples of periodic trichotomy behavior

Abstract

We study the following kth order rational difference equation assuming nonnegative parameters and nonnegative initial conditions $$x_{n}=\frac{\alpha+\sum_{i=1}^{k}\beta_{i}x_{n-i}} {A+\sum_{j=1}^{k}B_{j}x_{n-j}},\quad n\in\mathbb{N}.$$ We develop a new periodic trichotomy result which unifies all currently known periodic trichotomy results regarding the above difference equation into three major families. The periodic trichotomy result presented in this article also contains as special cases some new examples of periodic trichotomy behavior.