Lately it has come more evident that there is a fast growing community of researchers working with topological tools in dynamical systems, signal theory and time series analysis. This is and attempt to put together resources in this field as i go through them.
Perea J.A. (2018) Topological Time Series Analysis
This paper is a short review to the way TDA, and in particular persistent homology, has been used for time series analysis.
Here is a list of selected references from the paper and the quote for why they were mentioned in the paper.
[...] measures of fractal geometry can also be derived from persistent homology
[19] R. MacPherson and B. Schweinhart. Measuring shape with topology. Journal of Mathematical Physics, 53(7):073516, 2012
[24] V. Robins. Towards computing homology from finite approximations. In Topology proceedings, volume 24, pages 503–532, 1999
[...] The underlying shape of [the sliding window embedding] can be used in inference, classification and learning tasks
[12] J. Garland, E. Bradley, and J. D. Meiss. Exploring the topology of dynamical reconstructions. Physica D: Nonlinear Phenomena, 334:49–59, 2016.
[30] B. Xu, C. J. Tralie, A. Antia, M. Lin, and J. A. Perea. Twisty takens: Ageometric characterization of good observations on dense trajectories.arXivpreprint arXiv:1809.07131
Applications
[4] J. J. Berwald, M. Gidea, and M. Vejdemo-Johansson. Automatic recognitionand tagging of topologically different regimes in dynamical systems.Disconti-nuity, Nonlinearity, and Complexity, 3(4):413–126, 2014.
[28] C. J. Tralie and M. Berger. Topological eulerian synthesis of slow motion peri-odic videos. In2018 25th IEEE International Conference on Image Processing(ICIP), pages 3573–3577, 2018.
[29] C. J. Tralie and J. A. Perea. (quasi) periodicity quantification in video data,using topology.SIAM Journal on Imaging Sciences, 11(2):1049–1077, 2018
[10] A. Dirafzoon, N. Lokare, and E. Lobaton. Action classification from motioncapture data using topological data analysis. InSignal and Information Pro-cessing (GlobalSIP), 2016 IEEE Global Conference on, pages 1260–1264. IEEE,2016 Saba Emrani, Thanos Gentimis and Hamid Krim (2014) IEEE Wheeze detection
Myers, Munch, Khasawneh (2019) Persistent homology of complex networks for dynamic state detection
An interesting new approach to dynamic state detection using persistent homology with ordinal networks and Takens' embedding for dynamyc state detection. The comparison with the standard Lyapunov exponent is interesting and might be interesting to be further developed.
Brain: this paper has example applications to EEG and ECG
Gholizadeh S., Zadrozny W. (2018) A Short Survey of Topological Data Analysis in Time Series and Systems Analysis
Konstantin Mischaikow
the old school of standard topology and dynamical systems
Vector Field Editing and Periodic Orbit Extraction Using Morse Decomposition
Liz Bradley
Dynamic and TDA -- more mathsy PHETS code: Persistent Homology of Embedded Time series
Exploring the topology of dynamical reconstruction
Jose' Perea
Video and quasi-periodicity
C. J. Tralie
time series embedding -- using the song approach for EEG dynamic?
Marco Pettini
has some papers with Vaccarino about detection of ohase states and transitions in dynamical systems
Dynamics
A Look into Chaos Detection through Topological Data Analysis
HOMOLOGY THEORY AND DYNAMICAL SYSTEMS(1974)
Using persistent homology to reveal hidden covariates in systems governed by the kinetic Ising model
Signal theory
Topological Signal Processing overSimplicial Complexes
Topological data analysis for true step detection in periodic piecewise constant signals
Useful(?) Theory
The Homological Nature of Entropy
BioInformatics
Robust Detection of Periodic Patterns in GeneExpression Microarray Data using Topological Signal Analysis period detection for gene data
Topological methods for genomics has an example with phase space of a pendulum
Gauging functional brain activity: from distinguishability to accessibility Theory behind distinguishability of fMRI signal