We present a Lohner-type algorithm for rigorous integration of systems of Delay Differential Equations (DDEs) with multiple delays and its application in computation of Poincar\'e maps to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase-space and it exploits the smoothing of solutions occurring in DDEs to produces enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behavior, for example, existence of (apparently) unstable periodic orbits in Mackey-Glass Equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the R\"ossler system).

翻译：我们提出了一种适用于含多个时滞的时滞微分方程（DDEs）系统严格积分的Lohner型算法，并将其应用于庞加莱映射的计算，以研究某些有界、永恒解的动力学行为。该算法基于相空间中解的分段泰勒表示，利用DDEs中解的光滑化特性，生成高阶解包络。我们运用拓扑方法证明了多种动力学行为的存在性，例如：Mackey-Glass方程中（在数值上观察到混沌的参数区域）（表观）不稳定周期轨道的存在性，以及受时滞扰动的混沌常微分方程（Rössler系统）中符号动力学的持久性。