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PreprintCC BY 4.0arXiv

Complete Topological Invariant and Coding of Homoclinic Flows on the Two-Dimensional Sphere

  • Alexandr Prishlyak
  • Illia Ovtsynov
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Abstract

We study codimension one homoclinic flows on closed surfaces and introduce a complete topological invariant based on the structural graph of the flow and the distinguishing graphs of its atoms. This invariant completely determines the topological equivalence class of a homoclinic flow. For flows on the two-dimensional sphere, we construct a canonical coding algorithm that avoids the global renumbering required in the classical coding of Morse flows and substantially reduces the computational complexity. As an application, we derive recursive formulas for the number of topologically distinct homoclinic flows on the two-dimensional sphere and compute these numbers for flows with a small number of saddle singularities and closed orbits.

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Cite

BibTeXRIS
BibTeX
@misc{prishlyak2026complete,
  title = {{Complete Topological Invariant and Coding of Homoclinic Flows on the Two-Dimensional Sphere}},
  author = {Alexandr Prishlyak and Illia Ovtsynov},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02116},
  eprint = {2611.02116},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02116},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Complete Topological Invariant and Coding of Homoclinic Flows on the Two-Dimensional Sphere
AU  - Alexandr Prishlyak
AU  - Illia Ovtsynov
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.02116
UR  - https://arxiv.org/abs/2611.02116
AB  - We study codimension one homoclinic flows on closed surfaces and introduce a complete topological invariant based on the structural graph of the flow and the distinguishing graphs of its atoms. This invariant completely determines the topological equivalence class of a homoclinic flow. For flows on the two-dimensional sphere, we construct a canonical coding algorithm that avoids the global renumbering required in the classical coding of Morse flows and substantially reduces the computational complexity. As an application, we derive recursive formulas for the number of topologically distinct homoclinic flows on the two-dimensional sphere and compute these numbers for flows with a small number of saddle singularities and closed orbits.
ER  -