Recurrent signals give rise to trajectories that repeatedly return close to earlier states in state space. Many analysis methods therefore require a principled notion of similarity between states. In practice, a recurrence threshold sets the scale of the neighbourhood used to define when two states are considered close. Close returns can also support topology-preserving denoising in state space, aiming to reduce noise while preserving the trajectory's structure, which classical denoising methods may distort. The effectiveness of both denoising and recurrence analysis therefore depends critically on how these neighbourhoods are modelled and scaled. This work introduces a flow-aware ellipsoidal filtration for persistent homology based on a spatio--temporal covariance construction that estimates local flow geometry from both temporal and spatial neighbours. Unlike isotropic constructions based on balls (e.g.\ the Vietoris--Rips filtration), the proposed method assigns an ellipsoid to each point, with orientation and axis lengths determined by local flow variances. When a dominant $H_1$ feature reflects the recurrent loop structure, its persistence interval provides a data-driven scale selection. Across the considered experiments, flow-aware ellipsoidal neighbourhoods improve topology-preserving denoising and first-recurrence-time estimation relative to the Vietoris--Rips filtration. Overall, the results indicate that persistent homology can be more informative for dynamical systems when domain knowledge is used to incorporate anisotropy.

翻译：循环信号在状态空间中产生轨迹，这些轨迹会反复接近先前的状态。因此，许多分析方法需要对状态之间的相似性建立原则性定义。在实际应用中，递归阈值设定了用于判定两个状态是否接近的邻域尺度。接近的返回点还能支持状态空间中的拓扑保持降噪，旨在降低噪声的同时保持轨迹结构，而经典降噪方法可能会扭曲这种结构。降噪与递归分析的有效性因此关键取决于这些邻域如何建模与缩放。本文提出一种面向持续同调的流感知椭球过滤方法，其基于时空协方差构造，通过时间与空间邻域估计局部流动几何。与基于球体的各向同性构造（如Vietoris-Rips过滤）不同，所提方法为每个点分配一个椭球，其朝向与轴长由局部流动方差决定。当主导的$H_1$特征反映递归环结构时，其持续区间提供数据驱动的尺度选择。在实验案例中，相较于Vietoris-Rips过滤，流感知椭球邻域在拓扑保持降噪与首次递归时间估计上均有所改进。总体而言，结果表明当领域知识用于融入各向异性时，持续同调对动力系统能提供更具信息量的分析。