In this work we begin a theoretical and numerical investigation on the spectra of evolution operators of neutral renewal equations, with the stability of equilibria and periodic orbits in mind. We start from the simplest form of linear periodic equation with one discrete delay and fully characterize the spectrum of its monodromy operator. We perform numerical experiments discretizing the evolution operators via pseudospectral collocation, confirming the theoretical results and giving perspectives on the generalization to systems and to multiple delays. Although we do not attempt to perform a rigorous numerical analysis of the method, we give some considerations on a possible approach to the problem.

翻译：本文从理论和数值角度研究中立型更新方程演化算子的谱，重点关注平衡点与周期轨的稳定性。我们从含单一离散时滞的最简线性周期方程出发，完整刻画了其单值算子谱的特征。通过伪谱配置法对演化算子进行离散化数值实验，验证了理论结果，并为推广至系统及多时滞情形提供了展望。尽管未对该方法进行严格的数值分析，我们仍对问题的可能求解思路进行了若干探讨。