Logistic Map Potentials

TL;DR

This paper introduces methods to compute and analyze the underlying single particle potentials of the logistic map, revealing complex branch structures and transformations for different parameter values.

Contribution

It presents a novel approach to derive all branch potentials of the logistic map using functional transformations, including cases with infinite branch points.

Findings

Successfully computed potentials for s=5/2 and s=10/3

Demonstrated the presence of infinite branch points for s>2

Showed how switchback potentials relate through transformations

Abstract

We develop and illustrate methods to compute all single particle potentials that underlie the logistic map, x --> sx(1-x) for 0<s<=4. We show that the switchback potentials can be obtained from the primary potential through functional transformations. We are thereby able to produce the various branches of the corresponding analytic potential functions, which have an infinite number of branch points for generic s>2. We illustrate the methods numerically for the cases s=5/2 and s=10/3.