Discretizing hyperbolic periodic orbits of delay differential equations

Abstract

In this paper we generalize the results on discretizing exponentially stable periodic orbits of delay differential equations obtained in [9] to the case of unstable hyperbolic periodic orbits. Moreover, we prove persistence of local invariant manifolds around the periodic orbit and a shadowing theorem which allows us to compare the dynamics of the original solution operator and its discretization in a neighborhood of the periodic orbit. The proofs use ideas from [1] but our assumptions on the closeness of the solution operator and its discretization — motivated by concrete numerical applications — are weaker. To overcome the difficulty caused by this weaker closeness assumption we use the smoothing effect of delay equations.