A Look into Chaos Detection through Topological Data Analysis

Abstract: Traditionally, computation of Lyapunov exponents has been the marque method for identifying chaos in a time series. Recently, new methods have emerged for systems with both known and unknown models to produce a definitive 0--1 diagnostic. However, there still lacks a method which can reliably perform an evaluation for noisy time series with no known model. In this paper, we present a new chaos detection method which utilizes tools from topological data analysis. Bi-variate density estimates of the randomly projected time series in the $p$-$q$ plane described in Gottwald and Melbourne's approach for 0--1 detection are used to generate a gray-scale image. We show that simple statistical summaries of the 0D sub-level set persistence of the images can elucidate whether or not the underlying time series is chaotic. Case studies on the Lorenz and Rossler attractors as well as the Logistic Map are used to validate this claim. We demonstrate that our test is comparable to the 0--1 correlation test for clean time series and that it is able to distinguish between periodic and chaotic dynamics even at high noise-levels. However, we show that neither our persistence based test nor the 0--1 test converge for trajectories with partially predicable chaos, i.e. trajectories with a cross-distance scaling exponent of zero and a non-zero cross correlation.