Representation of period doubling by digraphs and characteristic polynomials

TL;DR

This paper explores how digraphs and characteristic polynomials can represent period doubling in nonlinear dynamical systems, providing a structured approach to understanding the transition to chaos.

Contribution

It introduces a novel method linking digraphs and polynomials to analyze period doubling, enhancing the theoretical framework for dynamical systems.

Findings

Established a partial ordering of cyclic permutations via digraphs

Applied the method to analyze period doubling in nonlinear systems

Provided insights into the route to chaos through polynomial representations

Abstract

A general procedure which defines a partial ordering of cyclic permutations induced by continuous maps is known for constructing immediate successors to a cycle. We expound on this procedure in terms of labelled digraphs and characteristic polynomials then apply this study to period doubling, the most common route to chaos for a nonlinear dynamical system.