  Changes in the parameters of dynamical systems can cause the state of the
system to shift between different qualitative regimes. These shifts, known as
bifurcations, are critical to study as they can indicate when the system is
about to undergo harmful changes in its behavior. In stochastic dynamical
systems, there is particular interest in P-type (phenomenological)
bifurcations, which can include transitions from a mono-stable state to
multi-stable states, the appearance of stochastic limit cycles, and other
features in the probability density function (PDF) of the system's state.
Current practices are limited to systems with small state spaces, cannot detect
all possible behaviours of the PDFs, and mandate human intervention for
visually identifying the change in the PDF. In contrast, this study presents a
new approach based on Topological Data Analysis (TDA) that uses superlevel
persistence to mathematically quantify P-type bifurcations in stochastic
systems through a "homological bifurcation plot'' -- which shows the changing
ranks of 0th and 1st homology groups. Using these plots, we demonstrate the
successful detection of P-bifurcations on the stochastic Duffing,
Raleigh-Vander Pol and Quintic Oscillators given their analytical PDFs, and
elaborate on how to generate an estimated homological bifurcation plot given a
kernel density estimate (KDE) of these systems by employing a tool for finding
topological consistency between PDFs and KDEs.
