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Alexandr Prishlyak

Publications and source records attributed to Alexandr Prishlyak.

At least 19 recordsLinked to original sources
arXiv2026

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

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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arXiv2026

Enumeration of one-cycle signed chord diagrams and their duals for optimal Morse flows on closed surfaces

Optimal Morse vector fields on closed surfaces, i.e.\ fields with the minimal possible number of critical points (a single source and a single sink), are classified up to topological trajectory equivalence by one-cycle signed chord diagrams in Kybalko, Prishlyak and Shchurko (2018). We recall this construction and the associated duality, which corresponds to reversing the direction of the flow, in a unified encoding in which both endpoints of a chord carry the same label, and we make it algorithmic. We give an explicit traversal procedure that detects one-cycle diagrams, a canonical form of an equivalence class under cyclic shifts, reversal and relabeling of chords, and a construction of the dual diagram. On this basis we describe an algorithm that enumerates all one-cycle signed chord diagrams with a given number of chords, both for orientable and for non-orientable surfaces, groups them into equivalence classes, pairs each class with its dual, and identifies the self-dual ones. The algorithm is implemented in Python and archived in Zenodo; its output is validated against the Harer--Zagier numbers in the orientable case and against orbit-size counts in both cases. We also note that, when a one-cycle diagram is read as a cellular embedding of a one-vertex graph, its dual diagram describes the embedding of the dual graph.

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arXiv2026

The structure of Morse flows and co-dimension one gradient flows on the sphere with holes

We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix diagrams of flows. The saddle-node singularity is specified by selecting a separatrix in the diagram of the flow befor the bifurcation and the saddle connection is specified by a separatrix, which conect two saddles.

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arXiv2025

Algorithms and topological invariants for dynamic systems. III. Algorithms for Recognition and Classification of 2-Dimensional Surfaces

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 (arXiv:2502.00506 ) we discused the main discrete topological structures used in the topological theory of dynamic systems. In third part we construct algorithms that allow us recognise 2-manifolds and 3-manifolds in simplicial complexes and regular CW-complex and detreminate topological type for 2-manifolds (connectivity, type of orientability, genus, number of boundary component).

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arXiv2025

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

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.

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arXiv2025

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

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.

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arXiv2024

Structure of optimal gradient flows bifurcations on closed surfaces

We consider structure of typical gradient flows bifurcations on closed surfaces with minimal number of singular points. There are two type of such bifurcations: saddle-node (SN) and saddle connections (SC). The structure of a bifurcation is determinated by codimension one flow in the moment of bifurcation. We use the chord diagrams to specify the flows up to topological trajectory equivivalence. A chord diagram with a marked arc is complete topological invariant of a SN-bifurcations and a chord diagram with T-insert -- of SC-bifurcations. We list all such diagrams for flows on norientable surfaces of genus at most 2 and nonorientable surfaces of genus at most 3. For each of diagram we found inverse one that correspond the inverse flow.

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arXiv2024

Structure of the codimesion one gradient flows with at most six singular points on the Möbius strip

We describe all possible topological structures of Morse flows and typical one-parametric gradient bifurcation on the Möbius strip in the case that the number of singular point of flows is at most six. To describe structures, we use the separatrix diagrams of flows. The saddle-node bifurcation is specified by selecting a separatrix in the diagram of the Morse flow befor the bifurcation and the saddle connection is specified by a separatrix, which connect two saddles on the diagram.

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arXiv2024

Structure of Morse flows with at most six singular points on the torus with a hole

We describe all possible topological structures of Morse flows and typical gradient saddle-nod bifurcation of flows on the 2-dimensional torus with a hole in the case that the number of singular point of flows is at most six. To describe structures, we use separatrix diagrams of flows. The saddle-node bifurcation is specified by selecting a separatrix in the separatrix diagram of the flow befor the bifurcation.

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arXiv2023

Morse functions with four critical points on immersed 2-spheres

We investigate topological properties of simple Morse functions with 4 critical points on immersed 2-spheres. To classify such functions, dual graph of immersion and Reeb graphs is used. We have found all possible structures of the functions:6 structures with 4 critical points on one 1-strata component, 7 structures with two points on the 1-strata and two points on the 2-strata, 7 structures with two 1-stratas and a three-connected 2-strata, three structures with two 1-stratas and without a three-connected 2-strata.

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arXiv2023

Structures of the flows with a unique singular point on the 2-dimensional disk

We investigate topological propeties of flows with one singular point and without closed orbits on the 2-dimensional disk. To classify such flows, destingueshed graph is used, which is a two-colored rooted tree imbedded in the plane. We construct a code of the flow and have found all possible structures of the flows with no more then 7 sepapratrices.

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arXiv2023

Typical one-parameter bifurcations of gradient flows with at most six singular points on the 2-sphere with holes

We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix diagrams of flows. The saddle-node singularity is specified by selecting a separatrix in the diagram of the flow befor the bifurcation and the saddle connection is specified by a separatrix, which conect two saddles.

math.DS↗
arXiv2023

Gradient vector fields of codimension one on the 2-sphere with at most ten singular points

We describe all possible topological structures of codimension one gradient vector fields on the shpere with at most ten singular points. To describe structures, we use a graph whose edges are one-dimensional stable manifolds. The saddle-node singularity is specified by selecting a pair of vertices-edge or edge-face, and the saddle connection is specified by a T-vertex.

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arXiv2023

Topological structure of Morse functions on the projective plane

To investigate the topological structure of Morse functions on the projective plane we use the Reeb graphs. We describe it properties and prove that it is a complete topological invariant of simple Morse function on $\mathbb{R} P^2$. We prove recurent formulas for the number of Reeb graphs with the given number of sadles (vertex).

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arXiv2022

Visualization of Morse flow with two saddles on 3-sphere diagrams

We describe all possible topological structures of Morse-Smale flows without closed trajectories on a three-dimensional sphere, which have two sources, two sinks, one saddle of Morse index 1, one saddle of Morse index 2, and no more than 10 saddle connections. To classify such flows, a generalized Heegaard diagram or Pr-diagram is used, which in this case consists of a sphere and two closed curves, the intersection points of which correspond to saddle connections. We have found all possible, up to homeomorphism, ways to embed two circles in a 2-sphere with no more than 10 points of transversal intersection and construct its planar visualisations.

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arXiv2022

Three-color graph as the 1-skeleton of the 2-sphere triangulation

The paper is devoted to finding the colorings of the edges of the 1-skeleton of triangulations of the 2-sphere in three colors so that for each face all three of its sides have different colors. First, by the method of adding one vertex inside the triangle or on its side, we enumerate all tiangulations with no more than 8 vertices. Next, one triangulation with 6 and 7 vertices, each with two different colors, was found. And finally, it is shown that other triangulations, which have less than 8 vertices, have one coloring each.

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arXiv2022

Morse flows with fixed points on the boundary of 3-manifold

The paper is devoted to the study of topological properties, structure and classification of Morse flows with fixed points on the boundary of three-dimensional manifolds. We construct a complete topological invariant of a Morse flow, Pr-diagram, which is similar to the Heegaard diagram of a closed 3-manifold.

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