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：本文从理论与数值角度研究中立型更新方程演化算子的谱，旨在探讨平衡态与周期轨道的稳定性。我们从含单个离散时滞的最简线性周期方程入手，完整刻画其单值算子的谱。通过伪谱配置法离散演化算子进行数值实验，验证了理论结果，并探讨了向系统及多时滞情形的推广。尽管未对该方法进行严格的数值分析，我们仍就可能的处理方式给出了一些思考。