Generalized Logistic Maps and Convergence

TL;DR

This paper explores three cubic recurrence relations, generalizing classic logistic maps, and investigates their asymptotic series, revealing intrinsic constants linked to initial conditions and the functions' behaviors.

Contribution

Introduces three generalized cubic recurrences extending logistic maps and analyzes their asymptotic series with intrinsic constants.

Findings

Identification of asymptotic series with initial-condition-dependent constants

Extension of logistic map theory to cubic recurrences

Insights into the intrinsic properties of generalized maps

Abstract

We treat three cubic recurrences, two of which generalize the famous iterated map $x \mapsto x (1-x)$ from discrete chaos theory. A feature of each asymptotic series developed here is a constant, dependent on the initial condition but otherwise intrinsic to the function at hand.