Lattice Structures for Attractors I

TL;DR

This paper explores the algebraic lattice structures underlying attractors and repellers in dynamical systems, aiming to develop computational algorithms for their manipulation.

Contribution

It introduces a formal algebraic framework for analyzing attractors and repellers using distributive lattices, bridging algebraic and topological methods.

Findings

Lattice structures effectively model attractors and repellers.

Separation of algebraic and topological properties facilitates algorithm development.

Foundation laid for computational manipulation of dynamical system structures.

Abstract

We describe the basic lattice structures of attractors and repellers in dynamical systems. The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and noninvertible. We separate those properties which rely solely on algebraic structures from those that require some topological arguments, in order to lay a foundation for the development of algorithms to manipulate these structures computationally.