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 Poincaré 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 produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, 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 Rössler system).

我们提出了一种 Lohner 型算法，用于严格集成具有多个延迟的延迟微分方程 (DDE) 系统，及其在 Poincaré 映射计算中的应用，以研究一些有界、永恒解的动力学。该算法基于相空间中解的分段泰勒表示，并利用 DDE 中出现的解的平滑来产生高阶解的包围。我们应用拓扑技术来证明各种动力学行为，例如，在 Mackey-Glass 方程中（在数值上观察到混沌的参数范围内）存在（显然）不稳定的周期轨道，以及延迟符号动力学的持久性 -扰动混沌 ODE（Rössler 系统）。