Algorithms and topological invariants for dynamic systems. II. Discrete Structures

TL;DR

This paper develops algorithms and invariants based on discrete topological structures to classify surfaces, functions, and vector fields, advancing the topological analysis of dynamic systems.

Contribution

It introduces new algorithms and invariants utilizing discrete structures like simplicial complexes and Morse theory for classifying topological types in dynamic systems.

Findings

Algorithms for distinguishing topological types of surfaces and vector fields.

Use of discrete Morse functions and homology groups for topological classification.

Application of Poincare rotation index in discrete settings.

Abstract

We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused basic concepts of diferential topology. In the second part we discus the main discrete topological structures used in the topological theory of dynamic systems: simplicial complexes, regular SW-complexes, Euler characteristic and homology groops, Morse-Smale complexes and handle decomposition of manifolds, Poincare rotation index of vector field, discrete Morse function and vector fields.