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Mrozek, Computational Homology ( Springer, 2004).

Bradley, “Computational topology at multiple resolutions: Foundations and applications to fractals and dynamics,” Ph.D. Robinson, Topological Signal Processing ( Springer, 2014), Vol. Halverson, “ Topological data analysis of biological aggregation models,” PLoS One 10, e0126383 (2015).

The results show that our approach reveals more information about the shape of the periodic attractor than standard tools, and it has more favorable computational time in comparison with the Rösenstein algorithm for computing the maximum Lyapunov exponent. We further prove the relationship between the Wasserstein distance to the empty diagram and the norm of the Betti vector, which shows that an even more simplified version of the information has the potential to provide insight into the bifurcation parameter. We validate our methods using numerical experiments on ten dynamical systems and contrast the results with existing tools that use the maximum Lyapunov exponent. The specific bifurcations we investigate are transitions from periodic to chaotic behavior or vice versa in a one-parameter collection of differential equations. Betti numbers are topological invariants of topological spaces, while the CROCKER plot is a coarsened but easy to visualize data representation of a one-parameter varying family of persistence barcodes.

Therefore, we describe an alternative method based on persistent homology-a tool from topological data analysis-that utilizes Betti numbers and CROCKER plots. Existing tools for bifurcation detection from signals of dynamical systems typically are either limited to a special class of systems or they require carefully chosen input parameters and a significant expertise to interpret the results.

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