Rigorous computation of expansion in one-dimensional dynamics

TL;DR

This paper presents a rigorous, computer-assisted algorithm for computing lower bounds on uniform expansion in one-dimensional dynamical systems, demonstrated on quadratic maps.

Contribution

The authors develop an effective, interval arithmetic-based method with graph algorithms for rigorous bounds, and provide a publicly available software implementation.

Findings

Successfully computes lower bounds for expansion in quadratic maps

Provides a rigorous numerical approach with computer-assisted proof

Demonstrates effectiveness through application to quadratic family

Abstract

We introduce an effective algorithmic method for the computation of a lower bound for uniform expansion in one-dimensional dynamics. The approach employs interval arithmetic and thus provides a rigorous numerical result (computer-assisted proof). The method uses efficient graph algorithms and an iterative approach for optimal performance. A software implementation of the method is made publicly available. This is an example of a quantitative result in the theory of dynamical systems, as opposed to many qualitative results whose assumptions may be difficult to verify and the conclusions may have limited use in practical models that describe natural phenomena. We discuss and illustrate the effectiveness of our method and apply it to the quadratic map family.