Dynamics of Functions with an Eventual Negative Schwarzian Derivative

TL;DR

This paper generalizes the class of one-dimensional dynamical systems by considering functions whose some iterate has a negative Schwarzian derivative, extending known results to this broader class motivated by neuroscience applications.

Contribution

It introduces a new class of functions with an eventual negative Schwarzian derivative and demonstrates that many classical results extend to this class.

Findings

Many known dynamical results apply to functions with an iterate having negative Schwarzian derivative.

The class includes maps arising in neuroscience.

Generalization broadens the applicability of existing dynamical systems theory.

Abstract

In the study of one dimensional dynamical systems one often assumes that the functions involved have a negative Schwarzian derivative. In this paper we consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. The introduction of this class was motivated by some maps arising in neuroscience.