Persistence of Periodic Orbits under Statedependent Delayed Perturbations: Computerassisted Proofs
Abstract
A computerassisted argument is given, which provides existence proofs for periodic orbits in statedependent 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 statedependent delayed perturbation of the van der Pol equation.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.06391
 Bibcode:
 2021arXiv211106391G
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Numerical Analysis