Some properties of the simple nonlinear recursion $y(\ell + 1) = [1-y(\ell)]^p$ with $p$ an arbitrary positive integer

TL;DR

This paper analyzes the behavior of solutions to a simple nonlinear recursion involving an arbitrary positive integer power, providing insights into its properties and solution characteristics.

Contribution

It presents an analysis of the properties and behavior of solutions to a specific nonlinear recursion for any positive integer power.

Findings

Solutions' behavior is characterized for different p values.

The recursion exhibits predictable dynamics based on initial conditions.

Properties of fixed points and stability are discussed.

Abstract

It is shown that the behavior of the solutions of the nonlinear recursion $y(\ell + 1) = [1-y(\ell)]^p$ -- where the dependent variable $y(l)$ is a real number, $\ell= 0; 1; 2...$ is the independent variable, and $p$ is an arbitrary positive integer -- is easily ascertainable.