Period doubling, information entropy, and estimates for Feigenbaum's constants

TL;DR

This paper introduces a novel approach combining information entropy and symbolic dynamics to define period doubling and provides simple estimates for Feigenbaum's constants related to log 2 and the golden ratio.

Contribution

It presents a new method to analyze period doubling using information entropy, leading to unexpected simple estimates for Feigenbaum's constants.

Findings

Feigenbaum's constants estimated using entropy and symbolic dynamics

Constants relate to log 2 and the golden ratio

New insights into universal scaling in nonlinear systems

Abstract

The relationship between period doubling bifurcations and Feigenbaum's constants has been studied for nearly 40 years and this relationship has helped uncover many fundamental aspects of universal scaling across multiple nonlinear dynamical systems. This paper will combine information entropy with symbolic dynamics to demonstrate how period doubling can be defined using these tools alone. In addition, the technique allows us to uncover some unexpected, simple estimates for Feigenbaum's constants which relate them to log 2 and the golden ratio, phi, as well as to each other.