A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems

Abstract

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.

Figures (12)

• Figure 1: Superlevel cubical persistence of image data. Figures correspond to (a) $K^{1.4}$, (b) $K^{1.05}$, (c) $K^{0.8}$ and (d) $K^{0.2}$.
• Figure 2: Sublevel persistence of a point cloud. Figures correspond to radii of (a) 0.14 (birth of smaller loop), (b) 0.31 (death of smaller loop), (c) 0.34 (birth of bigger loop), and (d) 0.73 (death of bigger loop).
• Figure 3: Evolution of the PDF for stochastic Duffing oscillator as $h$ changes its values. From left to right, $h$ = $-3, -1, 0, 1, 3$.
• Figure 4: Effect of bifurcation on the rank of $H_{0}$ for Duffing oscillator. From left to right, $h = -1, 0, 1$.
• Figure 5: Analytical homological bifurcation plots plots for Duffing Oscillator.