Period two implies chaos for a class of ODEs: a dynamical system approach

TL;DR

This paper discusses how period two solutions in certain scalar time-periodic differential equations imply chaotic behavior, using dynamical systems theory and the Bebutov flow to clarify the nature of chaos.

Contribution

It applies a dynamical systems approach to interpret chaos in scalar ODEs with periodic solutions, extending recent theorems to clarify the concept of chaos.

Findings

Presence of all periods in solutions indicates chaos.

Use of Bebutov flow clarifies the nature of chaos.

Period two solutions imply complex dynamics.

Abstract

The aim of this note is to set in the field of dynamical systems a recent theorem by Obersnel and Omari about the presence of periodic solutions of all periods for a class of scalar time-periodic first order differential equations without uniqueness, provided a subharmonic solution (and thus, for instance, a solution of period two) does exist. Indeed, making use of the Bebutov flow, we try to clarify in what sense the term "chaos" has to be understood and which dynamical features can be inferred for the system under analysis.