A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems

Sunia Tanweer, Firas A. Khasawneh, Elizabeth Munch, Joshua R. Tempelman

Published 2023-05-04Version 1

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 behaviour. In stochastic dynamical systems, particular interest is 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 distributions of the system's state space. Currently, the common practice is to visually analyze the probability density function to determine the type of state, but this approach is limited to experienced users, systems with small state spaces and mandate human intervention. In contrast, this study presents a new approach based on Topological Data Analysis (TDA) that uses the superlevel persistence of the probability or kernel density function to mathematically quantify P-type bifurcations in stochastic systems using the ranks of $0^{th}$ and $1^{st}$ homology groups.