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

This paper provides a unified framework connecting dynamical systems with tools from topological data analysis and geometric topology and inspires new interactions among dynamical systems, topology, and nonlinear analysis. To this end, we introduce a one-parameter family of ``chain recurrences'' that generalizes chain recurrence and induces a natural filtration on the underlying metric space of a dynamical system. In particular, the forward directions of the filtrations characterize the level of control required to return to the original position, and the backward directions capture the robustness of the recurrence. The resulting filtrations yield potentials and bifurcation diagrams of dynamical systems that encode the evolution of recurrent sets under bounded total or stepwise perturbations. In addition, we extend Morse graphs to one-parameter families of ``coarse Morse graphs,'' which evolve through vertex collapses reflecting coarse recurrence transitions. These constructions not only refine Conley's decomposition but also reveal singular limit behaviors as the perturbation level vanishes. Furthermore, we establish analogous filtrations for difference equations to bridge the theoretical framework with numerical analysis.