On the asymptotic stability of a rational multi-parameter first order difference equation

TL;DR

This paper investigates the long-term behavior of a rational multi-parameter first order difference equation, focusing on stability, equilibria, and 2-cycles, with implications for understanding complex discrete dynamical systems.

Contribution

It provides a detailed analysis of the equilibria and 2-cycles of a specific rational difference equation, including conditions for convergence and stability.

Findings

Identification of equilibrium points and their stability conditions.

Characterization of 2-cycles and their stability.

Conditions under which solutions converge to equilibria or 2-cycles.

Abstract

In this part we study the dynamics of the following rational multi-parameter first order difference equation x_{n+1} =(ax_{n}^3+ bx_{n}^2+cx_{n} + d)/x_{n}^3, x_{0}\in R^{+} where the parameters a, b, d together with the initial condition x_{0} are positive while the parameter c could accept some negative values. We investigate the equilibria and 2-cycles of this equation and analyze qualitative and asymptotic behavior of it's solutions such as convergence to an equilibrium or to a 2-cycle.