Numerical approximation of the non-essential spectrum of abstract delay differential equations

Abstract delay differential equations (ADDEs) extend delay differential equations (DDEs) from finite to infinite dimension. They arise in many application fields. From a dynamical system point of view, the stability analysis of an equilibrium is the first relevant question, which can be reduced to the stability of the zero solution of the corresponding linearized system. In the understanding of the linear case, the essential and the non-essential spectra of the infinitesimal generator are crucial. We propose to extend the infinitesimal generator approach developed for linear DDEs to approximate the non-essential spectrum of linear ADDEs. We complete the paper with the numerical results for a homogeneous neural field model with transmission delay of a single population of neurons.