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：我们推出一种Lohner型算法, 严格整合有多重延误的延迟差分(DDEs)系统, 并在计算Poincar\'e地图时应用该算法, 以研究某些捆绑的永久解决方案的动态。 该算法基于对阶段空间解决方案的简洁的Taylor表示, 并且利用 DDEs中出现的解决方案的平滑来生成高顺序解决方案的附文。 我们运用地形学技术来证明各种动态行为, 比如, 在Mackey-Glass Equation( 在从数字上观察混乱的参数体系中)存在( ) 不稳定的周期轨道, 以及符号动态在延缓干扰的混乱 ODE ( R\'ossler 系统) 中持续存在 。