Algorithms and topological invariants for dynamic systems. I. Basic definitions

TL;DR

This paper introduces algorithms and topological invariants to classify surfaces, functions, and vector fields based on their topological properties, focusing on the foundational structures in manifold topology.

Contribution

It develops new algorithms and invariants for distinguishing topological types of surfaces and dynamical systems, emphasizing fundamental structures in topology.

Findings

Algorithms for surface classification

Invariants for topological equivalence of functions and vector fields

Framework for analyzing topological structures in dynamical systems

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 we discus the main structures used in the topology of manifolds: vector fields, dynamical systems, Morse functions, cell decompositions, and the fundamental group.