Three-Color Graph of the Morse Flow on a Compact Surface with Boundary

Prishlyak, A. O.; Prus, A. A.

doi:10.1007/s10958-020-04964-1

Cited by 6 publications

(6 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For flows on a stratified set, their restriction to each two-dimensional stratum has the same structure as flows on a surface with boundary. This situation has been investigated in [30,31,34,35,37,38]. Structural stability of Morse-Smale flows was proved for closed manifolds in [20,42] and for manifolds with boundary in [23,40].…”

Section: Introductionmentioning

confidence: 99%

“…The use of graph theory to classify dynamical systems is typical in two and three dimensions. The topological classifications of flows were obtained on closed 2-manifolds in [7,16,19,20,21,38,1,27,30,42,28,29,36,26], and on 2-manifolds with the boundary in [5,8,17,30,27,24,35,21,36]. Complete topological invariants of Morse-Smale flows on 3-manifolds was constructed in [43,45,34,23,25,10,4,3].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Topological structure of optimal flows on the Girl's surface

Prishlyak

Loseva

2023

PIGC

View full text Add to dashboard Cite

We investigate the topological structure of flows on the Girl's surface which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of self-intersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and prove that there are unique images of the project plane in the form of a $2$-disk, in which the opposite points of the boundary are identified and this boundary belongs to the preimage of the $1$-skeleton of the surface. Second, we describe three structures of flows with one fixed point and no separatrices on the Girl's surface and prove that there are no other such flows. Third, we prove that Morse-Smale flows and they alone are structurally stable on the Boy's and Girl's surfaces. Fourth, we find all possible structures of optimal Morse-Smale flows on the Girl's surface. Fifth, we obtain a classification of Morse-Smale flows on the projective plane immersed on the Girl's surface. And finally, we describe the isotopic classes of these flows.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Topological structure of optimal flows on the Girl's surface

Prishlyak

Loseva

2023

PIGC

View full text Add to dashboard Cite

show abstract

“…It is convenient to use chord diagrams as complete topological invariants of polar Morse flows. Morse-Smale flows with singularities on a surface with boundary were investigated in [7,[12][13][14]. Since the trajectories of Hamiltonian vector field sgradf are contained in the level sets of f , in order to define such vector field (up to topological equivalence) we need to know only the direction on that vector field.…”

Section: Introductionmentioning

confidence: 99%

Flows with collective dynamics on a sphere

Prus¹,

Пришляк²,

Гурака³

2021

PIGC

View full text Add to dashboard Cite

show abstract

“…While considering flows on an annulus it is important to mention recent results on Morse flows on surfaces with a boundary [10].…”

Section: Introductionmentioning

confidence: 99%

On functional moduli of surface flows

Kruglov

Pochinka

Talanova

2020

PIGC

View full text Add to dashboard Cite

Currently, an complete topological classification has been obtained with respect to the topological equivalence of Morse-Smale flows, [9, 7], as well as their generalizations of Ω-stable flows on closed surfaces, [4]. Some results on topological conjugacy classification for such systems are also known. In particular, the coincidence of the classes of topological equivalence and conjugacy of gradient-like flows (Morse-Smale flows without periodic orbits) was established in [3]. In the classical paper [8], it was proved that in the presence of connections (coincidence of saddle separatrices), the topological equivalence class of a Ω-stable flow splits into a continuum of topological conjugacy classes (has moduli). Obviously, each periodic orbit also generates at least one modulus associated with the period of that orbit. In the present work, it was established that the presence of a cell in a flow bounded by two limit cycles leads to the existence of an infinitely many stability moduli. In addition, a criterion for the topological conjugation of flows on such cells was found.

show abstract

Three-Color Graph of the Morse Flow on a Compact Surface with Boundary

Cited by 6 publications

References 12 publications

Topological structure of optimal flows on the Girl's surface

Topological structure of optimal flows on the Girl's surface

Flows with collective dynamics on a sphere

On functional moduli of surface flows

Contact Info

Product

Resources

About