标题： 粗链回归、具有有限误差的莫尔斯图以及循环的持久性 显示英文标题

标题： Coarse chain recurrence, Morse graphs with finite errors, and persistence of circulations

摘要： 本文发展了一种拓扑框架，用于描述有限能量摄动下的整体动力学，重点关注从吸引子基域的逃逸、停滞点的消除以及由有限总能量或逐步能量控制的周期行为的持久性。 为此，我们引入了链回归的广义概念，并为度量空间上的动力系统构建了相应的过滤器。 这些过滤器细化了康利的经典理论，并能够对有界能量输入下的动力学行为进行定量建模。 特别是，我们引入了“粗链回归”和带有有限误差的莫尔斯图的概念，这些概念产生了分支图，描绘了当扰动尺度增加时链回归组件如何演变。 我们的框架还揭示了奇异极限，即注入的总能量消失的情况。 该框架适用于映射和半流，提供了分析周期动力学鲁棒性、灵敏性和结构演化的工具。 应用包括流控制和在有限能量约束下维持循环的玩具模型。 显示英文摘要

摘要： This paper develops a topological framework to describe global dynamics under finite-energy perturbations, focusing on escapes from attracting basins, elimination of stagnation, and persistence of recurrent behaviors governed by finite total or stepwise energy. To this end, we introduce generalized concepts of chain recurrence and construct corresponding filtrations for dynamical systems on metric spaces. These filtrations refine Conley's classical theory and enable the quantitative modeling of dynamical behaviors under bounded energy inputs. In particular, we introduce concepts of "coarse chain recurrences" and Morse graphs with finite errors, which yield bifurcation diagrams depicting how chain recurrent components evolve as perturbation scales increase. Our framework also reveals singular limits, where the total injected energy vanishes. The framework applies to both maps and semiflows, providing tools to analyze robustness, sensitivity, and structural evolution of recurrent dynamics. Applications include toy models of flow control and circulation maintained under finite-energy constraints.