High-order Lohner-type algorithm for rigorous computation of Poincar\'e maps in systems of Delay Differential Equations with several delays

TL;DR

This paper introduces a high-order Lohner-type algorithm for the rigorous numerical integration of Delay Differential Equations with multiple delays, enabling precise computation of Poincaré maps to analyze complex dynamical behaviors.

Contribution

It develops a novel piecewise Taylor-based method tailored for DDEs with multiple delays, facilitating rigorous dynamical analysis and detection of complex behaviors.

Findings

Proved existence of unstable periodic orbits in Mackey-Glass Equation.

Confirmed persistence of symbolic dynamics in delay-perturbed R"ossler system.

Achieved high-order enclosures of solutions in DDEs using the new algorithm.

Abstract

We present a Lohner-type algorithm for rigorous integration of systems of Delay Differential Equations (DDEs) with multiple delays and its application in computation of Poincar\'e maps to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase-space and it exploits the smoothing of solutions occurring in DDEs to produces enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behavior, for example, existence of (apparently) unstable periodic orbits in Mackey-Glass Equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the R\"ossler system).