Filtrations Indexed by Attracting Levels and their Applications

TL;DR

This paper introduces a new class of filtrations based on attracting levels in dynamical systems, enhancing topological data analysis by quantifying trajectory sensitivities and attraction breakdowns.

Contribution

It presents a novel filtration framework applicable to metric and partial maps, connecting dynamical systems with algebraic topology and demonstrating practical applications.

Findings

Filtrations identify regions of heightened sensitivity in cyclone forecasts.

Framework applies to general partial maps with cost functions.

Demonstrates potential for topological data analysis in dynamical systems.

Abstract

We introduce a new class of filtrations indexed by attracting levels in dynamical systems, providing novel inputs for persistent homology and related methods in topological data analysis. These filtrations quantify, in a forward direction, the sensitivity of trajectories with respect to attractors under perturbations and, in a backward direction, the perturbation magnitude at which attraction breaks down. The construction applies not only to maps on metric spaces but also to general partial maps with cost functions, yielding a filtration-theoretic framework with connections to algebraic topology. This generality ensures complementary filtrations when terminal states are good or bad, inducing natural decompositions of the underlying space. As an illustration, we apply the framework to ensemble forecasts of tropical cyclones, where the filtrations identify regions of heightened sensitivity, demonstrating the potential of our approach as a new tool for topological data analysis applied to dynamical systems.