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

Alexandr Prishlyak

TL;DR

This work develops algorithms and discrete topological invariants to classify surfaces and dynamical-system structures up to topological equivalence in low dimensions. It advances from classical simplicial complexes to regular cell complexes, then defines Euler characteristics and homology, followed by Morse–Smale complexes and handle decompositions, and culminates with Poincaré–Hopf indices and discrete Morse theory. The contributions include formal definitions, concrete RCC and simplicial constructions (e.g., torus RCCs, S^3 decompositions), and computational frameworks (discrete Morse functions, gradient fields) to derive homology and invariants via finite cell data. The methods enable efficient, computer-amenable recognition of manifolds and dynamical structures on them, with explicit guidance for calculating invariants like $\chi$, $H_i$, and $i_p(X)$ in low dimensions. Overall, the paper lays a cohesive toolkit for discrete topological analysis of surfaces and vector fields, linking combinatorial models to classical theorems such as Poincaré–Hopf and Morse theory, with practical implications for topology-based classification and computation.

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.

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

TL;DR

This work develops algorithms and discrete topological invariants to classify surfaces and dynamical-system structures up to topological equivalence in low dimensions. It advances from classical simplicial complexes to regular cell complexes, then defines Euler characteristics and homology, followed by Morse–Smale complexes and handle decompositions, and culminates with Poincaré–Hopf indices and discrete Morse theory. The contributions include formal definitions, concrete RCC and simplicial constructions (e.g., torus RCCs, S^3 decompositions), and computational frameworks (discrete Morse functions, gradient fields) to derive homology and invariants via finite cell data. The methods enable efficient, computer-amenable recognition of manifolds and dynamical structures on them, with explicit guidance for calculating invariants like , , and in low dimensions. Overall, the paper lays a cohesive toolkit for discrete topological analysis of surfaces and vector fields, linking combinatorial models to classical theorems such as Poincaré–Hopf and Morse theory, with practical implications for topology-based classification and computation.

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.

Paper Structure

This paper contains 7 sections, 10 theorems, 38 equations, 8 figures.

Table of Contents

  1. Simplicial Complexes
  2. Regular Cell Complexes
  3. Euler Characteristic and Homology Groups
  4. Morse–Smale Complex and Handle Decomposition
  5. Poincaré–Hopf Theorem on the Indices of Vector Fields
  6. Discrete Morse Functions and Fields
  7. Literature Review

Key Result

Theorem 1

(about simplicial approximation). If there is a continuous mapping of polyhedra, then there exists such a subdivision $L$ of the complex $K$ and a simplicial mapping $L$ to $M$, which is homotopic to $f$.

Figures (8)

  • Figure 1: RCC of the 2-dimensional torus
  • Figure 2: Gradient flow with saddle connection on the torus
  • Figure 3: Bringing stable and unstable manifolds into general position
  • Figure 4: Examples of rotation indices
  • Figure 5: Reduction of two points of tangency
  • ...and 3 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1
  • Theorem 5
  • Theorem 6
  • Lemma 1
  • Lemma 2
  • Lemma 3