Exploring Topological Data Analysis for Identifying Phenomenological
Stochastic Bifurcations
Sunia Tanweer, Firas Khasawneh, Elizabeth Munch, Joshua Tempelman
Michigan State University
::
Dept. of Mechanical Engineering
May 15, 2023
SIAM 2023
Portland, OR
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 1 / 18
Contributors
Firas A. Khasawneh
1
Elizabeth Munch
1
Joshua Tempelman
1, 2
This material is based upon work supported by the Air Force Office of
Scientific Research under award number FA9550-22-1-0007.
1
Michigan State University.
2
University of Illinois-Urbana Champaign.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 2 / 18
Motivation
Population Growth
1
Coupled Chemical Reactions
Freeway Traffic
Aeroelastic Systems
Neuronal Models
Financial Markets
*All images generated through craiyon.com (AI text to image generator)
1
David G. Kendall. “Stochastic Processes and Population Growth”. In: Journal of the Royal Statistical Society: Series B (Methodological) 11.2 (July 1949),
pp. 230–264. doi: 10.1111/j.2517-6161.1949.tb00032.x. url: https://doi.org/10.1111/j.2517-6161.1949.tb00032.x.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 3 / 18
Motivation
Population Growth
Coupled Chemical Reactions
2
Freeway Traffic
Aeroelastic Systems
Neuronal Models
Financial Markets
*All images generated through craiyon.com (AI text to image generator)
2
Xiaodong Cai. “Exact stochastic simulation of coupled chemical reactions with delays”. In: The Journal of Chemical Physics 126.12 (Mar. 2007), p. 124108.
doi: 10.1063/1.2710253. url: https://doi.org/10.1063/1.2710253.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 3 / 18
Motivation
Population Growth
Coupled Chemical Reactions
Freeway Traffic
3
Aeroelastic Systems
Neuronal Models
Financial Markets
*All images generated through craiyon.com (AI text to image generator)
3
Kai Nagel and Michael Schreckenberg. “A cellular automaton model for freeway traffic”. In: Journal de Physique I 2.12 (Dec. 1992), pp. 2221–2229. doi:
10.1051/jp1:1992277. url: https://doi.org/10.1051/jp1:1992277.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 3 / 18
Motivation
Population Growth
Coupled Chemical Reactions
Freeway Traffic
Aeroelastic Systems
4
Neuronal Models
Financial Markets
*All images generated through craiyon.com (AI text to image generator)
4
J. Venkatramani, Sunetra Sarkar, and Sayan Gupta. “Intermittency in pitch-plunge aeroelastic systems explained through stochastic bifurcations”. In:
Nonlinear Dynamics 92.3 (Feb. 2018), pp. 1225–1241. doi: 10.1007/s11071-018-4121-5. url: https://doi.org/10.1007/s11071-018- 4121-5.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 3 / 18
Motivation
Population Growth
Coupled Chemical Reactions
Freeway Traffic
Aeroelastic Systems
Neuronal Models
5
Financial Markets
*All images generated through craiyon.com (AI text to image generator)
5
Seiji Tanabe and K. Pakdaman. “Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations”. In: Physical Review E 63.3 (Feb.
2001). doi: 10.1103/physreve.63.031911. url: https://doi.org/10.1103/physreve.63.031911.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 3 / 18
Motivation
Population Growth
Coupled Chemical Reactions
Freeway Traffic
Aeroelastic Systems
Neuronal Models
Financial Markets
6
*All images generated through craiyon.com (AI text to image generator)
6
Carl Chiarella et al. “The stochastic bifurcation behaviour of speculative financial markets”. In: Physica A: Statistical Mechanics and its Applications 387.15
(June 2008), pp. 3837–3846. doi: 10.1016/j.physa.2008.01.078. url: https://doi.org/10.1016/j.physa.2008.01.078.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 3 / 18
Persistent Homology
What is Persistent Homology?
A way to watch how the homology of a filtration (sequence) of topological spaces changes so
that we can understand something about the space.
Given topological space K and filtration
K
0
⊆ K
1
⊆ K
2
⊆ · · · ⊆ K
n
gives a sequence of maps on homology
H
1
(K
0
)
//
H
1
(K
1
)
//
H
1
(K
2
)
//
· · ·
//
H
1
(K
n
)
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 8 / 18
Persistent Homology
What is Persistent Homology?
A way to watch how the homology of a filtration (sequence) of topological spaces changes so
that we can understand something about the space.
Given topological space K and filtration
K
0
⊆ K
1
⊆ K
2
⊆ · · · ⊆ K
n
gives a sequence of maps on homology
H
1
(K
0
)
//
H
1
(K
1
)
//
H
1
(K
2
)
//
· · ·
//
H
1
(K
n
)
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 8 / 18
Persistent Homology
What is Persistent Homology?
A way to watch how the homology of a filtration (sequence) of topological spaces changes so
that we can understand something about the space.
Given topological space K and filtration
K
0
⊆ K
1
⊆ K
2
⊆ · · · ⊆ K
n
gives a sequence of maps on homology
H
1
(K
0
)
//
H
1
(K
1
)
//
H
1
(K
2
)
//
· · ·
//
H
1
(K
n
)
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 8 / 18
Topological Consistency in Density Estimates
1
L − ϵ superlevel set
Cycles = 2
L + ϵ superlevel set
Cycles = 1
Intersection
Cycles = 1
1
Omer Bobrowski, Sayan Mukherjee, and Jonathan E. Taylor. “Topological consistency via kernel estimation”. In: Bernoulli 23.1 (Feb. 2017). doi:
10.3150/15-bej744. url: https://doi.org/10.3150/15-bej744.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 12 / 18
Topological Consistency in Density Estimates
1
L − ϵ superlevel set
Cycles = 2
L + ϵ superlevel set
Cycles = 1
Intersection
Cycles = 1
1
Omer Bobrowski, Sayan Mukherjee, and Jonathan E. Taylor. “Topological consistency via kernel estimation”. In: Bernoulli 23.1 (Feb. 2017). doi:
10.3150/15-bej744. url: https://doi.org/10.3150/15-bej744.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 12 / 18
Topological Consistency in Density Estimates
1
L − ϵ superlevel set
Cycles = 2
L + ϵ superlevel set
Cycles = 1
Intersection
Cycles = 1
1
Omer Bobrowski, Sayan Mukherjee, and Jonathan E. Taylor. “Topological consistency via kernel estimation”. In: Bernoulli 23.1 (Feb. 2017). doi:
10.3150/15-bej744. url: https://doi.org/10.3150/15-bej744.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 12 / 18
Topological Consistency in Density Estimates
1
L − ϵ superlevel set
Cycles = 2
L + ϵ superlevel set
Cycles = 1
Intersection
Cycles = 1
1
Omer Bobrowski, Sayan Mukherjee, and Jonathan E. Taylor. “Topological consistency via kernel estimation”. In: Bernoulli 23.1 (Feb. 2017). doi:
10.3150/15-bej744. url: https://doi.org/10.3150/15-bej744.
Sunia Tanweer (MSU) Bifurcation Detection May 15, 2023 - SIAM 12 / 18
