Persistence of Periodic Orbits under State-dependent Delayed Perturbations: Computer-assisted Proofs

TL;DR

This paper introduces a computer-assisted method to rigorously prove the existence of periodic orbits in state-dependent delayed perturbations of ODEs, combining validated numerics and polynomial inequalities.

Contribution

It develops a general algorithm that verifies polynomial inequalities to establish periodic orbits in perturbed delay equations, using Chebyshev series and validated numerics.

Findings

Proved existence of periodic orbits in a delayed van der Pol equation.

Developed an algorithm for verifying polynomial inequalities in delay perturbations.

Demonstrated the effectiveness of computer-assisted proofs in dynamical systems.

Abstract

A computer-assisted argument is given, which provides existence proofs for periodic orbits in state-dependent delayed perturbations of ordinary differential equations (ODEs). Assuming that the unperturbed ODE has an isolated periodic orbit, we introduce a set of polynomial inequalities whose successful verification leads to the existence of periodic orbits in the perturbed delay equation. We present a general algorithm, which describes a way of computing the coefficients of the polynomials and optimizing their variables so that the polynomial inequalities are satisfied. The algorithm uses the tools of validated numerics together with Chebyshev series expansion to obtain the periodic orbit of the ODE as well as the solution of the variational equations, which are both used to compute rigorously the coefficients of the polynomials. We apply our algorithm to prove the existence of periodic orbits in a state-dependent delayed perturbation of the van der Pol equation.