Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation

TL;DR

This paper introduces a rigorous algorithm for integrating Delay Differential Equations and applies it to prove the existence of specific periodic orbits in the Mackey-Glass equation using computer-assisted methods.

Contribution

The paper develops a new algorithm for the rigorous numerical integration of DDEs and demonstrates its effectiveness through a computer-assisted proof of periodic orbits in a classic biological model.

Findings

Proved existence of two attracting periodic orbits in the Mackey-Glass equation.

Validated the algorithm's capability for rigorous DDE integration.

Identified bifurcation points related to periodic orbits.

Abstract

We present an algorithm for the rigorous integration of Delay Differential Equations (DDEs) of the form $x'(t)=f(x(t-\tau),x(t))$. As an application, we give a computer assisted proof of the existence of two attracting periodic orbits (before and after the first period-doubling bifurcation) in the Mackey-Glass equation.