In this paper we focus on the computation of periodic solutions of Delay Differential Equations (DDEs) with constant delays. The method is based on defining a Poincaré section in a suitable functional space and looking for a fixed point of the flow in this section. This is done by applying a Newton method on a suitable discretisation of the section. To avoid computing and storing large matrices we use a GMRES method to solve the linear system because in this case GMRES converges very fast due to the compactness of the flow of the DDE. The derivatives of the Poincaré map are obtained in a simple way, by applying Automatic Differentiation to the numerical integration. The stability of the periodic orbit is also obtained in a very efficient way by means of Arnoldi methods. The examples considered include temporal and spatial Poincaré sections.

中文翻译：

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使用自动微分计算延迟微分方程的周期轨道

在本文中，我们专注于计算具有恒定延迟的时滞微分方程（DDE）的周期解。该方法基于在合适的功能空间中定义庞加莱部分，并在该部分中寻找流的固定点。这可以通过在截面的适当离散上应用牛顿法来完成。为了避免计算和存储大矩阵，我们使用GMRES方法求解线性系统，因为在这种情况下，由于DDE流程的紧凑性，GMRES收敛非常快。通过将自动微分应用于数值积分，可以以简单的方式获得庞加莱图的导数。还可以通过Arnoldi方法以非常有效的方式获得周期轨道的稳定性。所考虑的示例包括时间和空间庞加莱部分。