A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910318114897920 |
|---|---|
| author | Tanweer, Sunia Khasawneh, Firas A. Munch, Elizabeth Tempelman, Joshua R. |
| author_facet | Tanweer, Sunia Khasawneh, Firas A. Munch, Elizabeth Tempelman, Joshua R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03118 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems Tanweer, Sunia Khasawneh, Firas A. Munch, Elizabeth Tempelman, Joshua R. Dynamical Systems 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. |
| title | A Topological Framework for Identifying Phenomenological Bifurcations in Stochastic Dynamical Systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2305.03118 |