Algorithm for Rigorous Integration of Delay Differential Equations and the Computer-Assisted Proof of Periodic Orbits in the Mackey–Glass Equation

Abstract

We present an algorithm for the rigorous integration of delay differential equations (DDEs) of the form x′(t) = f(x(t- τ) , 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.