Chaos Detection with Persistent Homology
This data set contains MATLAB scripts which generate a Persistence Score (PS) for a time series. The time series are generated from dynamical systems (specifically the Lorenz model, Rossler model, and Logistics map) across an array of bifurcation parameters. The PS scores are generated from a persistence diagram which is taken of a bi-variate kernel density estimate of a planar projection of the time series data. This library contains MATLAB data structures which store the results several studies of the PS scores. Specifically, results are included which span a bifurcation of the Rossler, Logistics, and Lorenz models. The same results are given in separate data structures for time-series with added White Gaussian noise. The PS scores are computed along with 0/1 scores, and these scoring systems are compared. A MALTAB file (Scoring.m) unpacks and plots the results of this study. A Gaussian smoothing function is used in this study to condition gray scale images in preparation for persistent homology. The effect of th kernel width sigma on the PS scores is studied, and MATLAB data structures are included which contain the results of this study for clean data along with data contaminated with noise at an Signal to Noise Ratio of 30 dB over a span of sigma = 0.1 to sigma 4.0.
Steps to reproduce
A time-series is generated by simulating a dynamical system. This time series is then projected into a new planar space (See "On the Implementation of the 0-1 Test for Chaos", Gottwald and Melbourne, 2009). With these projections, the 0/1 test for chaos is applied via the correlation method. Alternatively, the data can be studied using topological data analysis. To do this, the planar projections are converted into gray scale images by taking the bi-variate kernel density estimate (see "Kernel density estimation via diffusion", Z. I. Botev, J. F. Grotowski, and D. P. Kroese 2010). A Gaussian smoothing function is applied to the gray scale images. Using the gray-scaled images, the DIPHA package (https://github.com/DIPHA/dipha) computes a persistence diagram for the image. In order to exectute the codes which generate new results based on persistence, the DIPHA package must be installed and compiled. The DIPHA executable must be moved to path as well. The 1D persistence data pulled from this, and persistence points very close to the origin are discarded. Several simple statistical summaries are taken of these persistence points to generate a score. These scores are then used to identify the shift points in the dynamical systems parameter space between regions of periodic and chaotic behavior.