Inferring symbolic dynamics of chaotic flows from persistence

Abstract

We introduce state space persistence analysis for deducing the symbolic dynamics of time-series data obtained from high-dimensional chaotic attractors. To this end, we adapt a topological data analysis technique known as persistent homology for the characterization of state space projections of chaotic trajectories and periodic orbits. By comparing the shapes along a chaotic trajectory to those of the periodic orbits, state space persistence analysis quantifies the geometric similarities of chaotic trajectory segments and the periodic orbits. We demonstrate the method by applying it to the three-dimensional Rössler system and a thirty-dimensional discretization of the Kuramoto-Sivashinsky partial differential equation in (1 + 1) dimensions.

A Topological Perspective on Weather Regimes

“…From a topological perspective, where no knowledge of fixed-points is implicit, these loops are what stand out as the major feature of CdV, implying that focusing attention on such features can highlight information which is otherwise being overlooked. This potential of topological methods to obtain efficient, simplified representations of chaotic dynamics was also noted in [30] using different ideas.…”

“…The potential of persistent homology as a tool for analysing dynamical systems was first suggested, among others, in [28], in which it was demonstrated that persistent homology can locate the holes of the Lorenz '63 system: see also [29]. Of particular relevance is the recent work of [30], which uses persistent homology and UPOs in order to obtain a simplified representation of chaotic dynamical systems, an approach similar in spirit to our paper. A recent application of persistent homology to the real atmosphere is [31], which studies 'atmospheric rivers' using a combination of homology and machine learning.…”

A topological perspective on weather regimes

“…From a topological perspective, where no knowledge of fixed-points is implicit, these loops are what stand out as the major feature of CdV, implying that focusing attention on such features can highlight information which is otherwise being overlooked. This potential of topological methods to obtain efficient, simplified representations of chaotic dynamics was also noted in [62] using different ideas.…”

“…The potential of topological techniques for dynamical systems analysis was first suggested, among others, in [34], in which it was demonstrated that persistent homology can locate the holes of the Lorenz '63 system. Of particular relevance is the recent work of [62], which uses persistent homology in order to obtain a simplified representation of chaotic dynamics, an approach similar in spirit to our paper. A recent application of persistent homology to the real atmosphere is [38], which studies 'atmospheric rivers' using a combination of homology and machine learning.…”

Spatial applications of topological data analysis: Cities, snowflakes, random structures, and spiders spinning under the influence

“…Moreover, the extension of homology to so-called "persistent homology" (PH) allows one to quantify holes in data in a meaningful way and has made it possible to apply homological ideas to a wide variety of empirical data sets [22,23]. PH is helpful for characterizing the "shape" of data, and the myriad applications of it include studies of protein structure [24][25][26][27], DNA structure [28], neuronal morphologies [29], computer vision [30], diurnal cycles in hurricanes [31], chaotic dynamics in differential equations [32], spatial percolation problems [33], and many others. Additionally, combining machine-learning approaches with PH has also been very useful for several classification problems [34][35][36][37].…”