Dynamics of a rational multi-parameter second order difference equation with cubic numerator and quadratic monomial denominator

TL;DR

This paper investigates the long-term behavior of solutions to a complex second-order rational difference equation with multiple parameters, focusing on convergence, divergence, and cyclical patterns.

Contribution

It provides a detailed analysis of the asymptotic dynamics of a multi-parameter rational difference equation with cubic and quadratic terms.

Findings

Conditions for convergence to equilibrium

Criteria for divergence to infinity

Existence of 2-cycle solutions

Abstract

The asymptotic behavior (such as convergence to an equilibrium, convergence to a 2-cycle, and divergence to infinity) of solutions of the following multi-parameter, rational, second order difference equation x_{n+1} =(ax_{n}^3+ bx_{n}^2x_{n-1}+cx_{n}x_{n-1}^2+dx_{n-1}^3)/x_{n}^2, x_{-1},x_{0}\in R, is studied in this paper.