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

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.

15 pages, 8 figures