A computational framework for connection matrix theory

TL;DR

This paper introduces a categorical, homological framework for connection matrix theory, transforming computational Conley theory into a homological approach for analyzing dynamical systems.

Contribution

It develops a categorical, homotopy-theoretic formulation of connection matrix theory and presents an algorithm based on algebraic-discrete Morse theory for computation.

Findings

Categorical framework for connection matrices established.

Algorithm for computing connection matrices developed.

Transforms computational Conley theory into a homological setting.

Abstract

The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, the ideas of Conley have been cast into a purely computational form. In this paper we introduce a computational, categorical framework for the connection matrix theory. This contribution transforms the computational Conley theory into a computational homological theory for dynamical systems. More specifically, within this paper we have two goals: 1) We cast the connection matrix theory into appropriate categorical, homotopy-theoretic language. We identify objects of the appropriate categories which correspond to connection matrices and may be computed within the computational Conley theory paradigm by using the technique of reductions. 2) We describe an algorithm for this computation based on algebraic-discrete Morse theory.