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，用于对具有多个延迟的时滞微分方程组 (DDE) 进行严格积分，并将其应用于庞加莱图的计算，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，并且利用 DDE 中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，麦基-格拉斯方程中（在数值观察到混沌的参数体系中）中（显然）不稳定周期轨道的存在以及延迟中符号动力学的持久性扰动混沌 ODE（罗斯勒系统）。