Dynamics Seminar

In this talk, we provide a unified framework that connects dynamical systems with tools from topological data analysis and geometric topology, and that inspires new interactions among dynamical systems, topology, and nonlinear analysis. To this end, we introduce one-parameter families of ''recurrences'' that generalize chain recurrence and the non-wandering property and induce natural filtrations on the underlying metric space of a dynamical system. At parameter zero, they coincide with the original concepts. Moreover, 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 coincide with the original Morse graph at parameter zero and evolve through vertex collapses reflecting coarse recurrence transitions.