Persistent homology, the flagship method from the field of Topological Data Analysis, is a powerful tool for measuring shape and structure of data. In this talk, we explore methods for using this tool to detect homological changes in the underlying structure of dynamical systems.  As a first step, we can simplify a vineyard of persistence diagrams into a CROCKER plot to provide visual representations of qualitative shifts in the structure of examples such as the Lorenz and Rossler systems. We can also construct a “homological bifurcation plot” to enable the identification of qualitative shifts, namely P-type (phenomenological) bifurcations, within stochastic dynamical systems, defined by structural changes in the probability density functions (PDF) of the state variables. The talk will explore the successful application of this method to stochastic oscillators, showcasing its effectiveness in algorithmically detecting P-bifurcations. This talk is based on joint work with many collaborators, including Firas Khasawneh, İsmail Güzel, Sunia Tanweer, Sarah Tymochko, Audun Myers, and David Muñoz.