Detecting bifurcations in dynamical systems with CROCKER plots

TL;DR

This paper introduces a topological data analysis method using CROCKER plots and Betti numbers to detect bifurcations in dynamical systems, offering improved insight and computational efficiency over traditional techniques.

Contribution

The authors develop a novel bifurcation detection approach based on persistent homology and CROCKER plots, applicable to various dynamical systems without extensive parameter tuning.

Findings

The method effectively detects bifurcations from periodic to chaotic behavior.

It provides richer information about attractor shapes than standard tools.

The approach is computationally faster than Lyapunov exponent calculations.

Abstract

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 significant expertise to interpret the results. Therefore, we describe an alternative method based on persistent homology -- a tool from Topological Data Analysis (TDA) -- that utilizes Betti numbers and CROCKER plots. 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. The specific bifurcations we investigate are transitions from periodic to chaotic behavior or vice versa in a one-parameter family of differential equations. We validate our methods using numerical experiments on ten dynamical systems and contrast the results with existing tools that use 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. 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 to the Rosenstein algorithm for computing the Lyapunov exponent from time series.