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

Alexandr Prishlyak

TL;DR

The paper addresses classification of surface topologies and dynamical systems through computable invariants and algorithms. It combines continuous and discrete viewpoints by leveraging Morse theory, cell complexes, and fundamental group methods to enable recognition and comparison of topological structures in dimensions up to $4$. Structured into four parts, it covers continuous structures (manifolds, vector fields, Morse functions, cell decompositions, and $\pi_1$), discrete specifications, algorithmic manifold recognition, and topological structures of functions and dynamics on low-dimensional manifolds. These developments aim to provide concrete, computable tools for identifying and differentiating topological types and dynamical behaviors in practical applications.

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.

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

TL;DR

The paper addresses classification of surface topologies and dynamical systems through computable invariants and algorithms. It combines continuous and discrete viewpoints by leveraging Morse theory, cell complexes, and fundamental group methods to enable recognition and comparison of topological structures in dimensions up to . Structured into four parts, it covers continuous structures (manifolds, vector fields, Morse functions, cell decompositions, and ), discrete specifications, algorithmic manifold recognition, and topological structures of functions and dynamics on low-dimensional manifolds. These developments aim to provide concrete, computable tools for identifying and differentiating topological types and dynamical behaviors in practical applications.

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.
Paper Structure (11 sections, 9 theorems, 60 equations, 13 figures)

This paper contains 11 sections, 9 theorems, 60 equations, 13 figures.

Table of Contents

  1. Basic Topological Definitions and Notations
  2. Topological Properties
  3. Topological Operations
  4. Classic Spaces and Their Notations
  5. Smooth Manifolds and Their Mappings
  6. Vector Fields and Dynamical Systems
  7. Morse Functions
  8. Cell Complexes
  9. Homotopic Mappings
  10. Fundamental Group
  11. Literature Review

Key Result

Theorem 1

(about the implicit function for submanifolds) Let a system of equations be given in the space $\mathbb{R}^n$ where $f_i$ are smooth functions and $M$ is the set of solutions to this system. If the rank of the matrix $J = \left( \frac{\partial f_i}{\partial x_j} \right)$ is equal to $k$ everywhere on the set $M$, then $M$ is a submanifold of dimension $n-k$.

Figures (13)

  • Figure 1: The Möbius strip $Mo$
  • Figure 2: The two-dimensional torus $T^2$
  • Figure 3: The Klein bottle $Kl$
  • Figure 4: coordinate mapping record
  • Figure 5: A mapping of a point neighbourhood
  • ...and 8 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8