Pseudospectral discretization of delay differential equations in sun-star formulation: Results and conjectures

In this paper we study the pseudospectral approximation of delay differential equations formulated as abstract differential equations in the $odot*$-space. This formalism also allows us to define rigorously the abstract variation-of-constants formula, where the $odot*$-shift operator plays a fundamental role. By applying the pseudospectral discretization technique we derive a system of ordinary differential equations, whose dynamics can be efficiently analyzed by existing bifurcation tools. To better understand to what extent the resulting finite-dimensional system ``mimics'' the dynamics of the original infinite-dimensional one, we study the pseudospectral approximations of the $odot*$-shift operator and of the $odot*$-generator in the supremum norm, which is the natural choice for delay differential equations, when the discretization parameter increases. In this context there are still open questions. We collect the most relevant results from the literature and we present some conjectures, supported by various numerical experiments, to illustrate the behavior w.r.t. the discretization parameter and to indicate the direction of ongoing and future research.