Lattice Structures for Attractors II

TL;DR

This paper investigates whether computational methods for attracting neighborhoods can fully capture the algebraic lattice structure of attractors in dynamical systems, enhancing understanding of global dynamics.

Contribution

It explores the extent to which existing computational algorithms can detect the complete algebraic lattice structure of attractors.

Findings

Computational methods can identify parts of the attractor lattice.

Some algebraic structures of attractors remain challenging to detect.

The study advances understanding of the relationship between computational detection and theoretical structure.

Abstract

The algebraic structure of the attractors in a dynamical system determine much of its global dynamics. The collection of all attractors has a natural lattice structure, and this structure can be detected through attracting neighborhoods, which can in principle be computed. Indeed, there has been much recent work on developing and implementing general computational algorithms for global dynamics, which are capable of computing attracting neighborhoods efficiently. Here we address the question of whether all of the algebraic structure of attractors can be captured by these methods.