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Circular Fleitas Scheme for Gradient-Like Flows on the Surface

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Abstract

In this paper, we obtain a classification of gradient-like flows on arbitrary surfaces by generalizing the circular Fleitas scheme. In 1975 he proved that such a scheme is a complete invariant of topological equivalence for polar flows on 2- and 3-manifolds. In this paper, we generalize the concept of a circular scheme to arbitrary gradient-like flows on surfaces. We prove that the isomorphism class of such schemes is a complete invariant of topological equivalence. We also solve exhaustively the realization problem by describing an abstract circular scheme and the process of realizing a gradient-like flow on the surface. In addition, we construct an efficient algorithm for distinguishing the isomorphism of circular schemes.

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Notes

  1. An invariant set \(A\subset M^{n}\) of a flow \(f^{t}:M^{n}\to M^{n}\) is called an attractor if it has a closed neighborhood \(U_{A}\), which is called trapping, such that \(f^{t}(U_{A})\subset\text{int}U_{A}\) for \(t>0\) and \(\bigcap\limits_{t>0}f^{t}(U_{A})=A\).

  2. Notice that an invariant similar to the circular scheme was used in [6] for a description of the connected components of gradient-like vector fields on closed surfaces.

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Funding

This work was supported by the Russian Science Foundation (Project No. 23-71-30008), except for Section 4, which was supported by the Laboratory of Dynamic Systems and Applications of the HSE, grant of the Ministry of Science and Higher Education of the Russian Federation, Agreement No. 075-15-2019-1931.

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Authors and Affiliations

  1. HSE University, ul. Bolshaya Pecherckaya 25/12, 603155, Nizhny Novgorod, Russia

    Vladislav D. Galkin, Elena V. Nozdrinova & Olga V. Pochinka

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  1. Vladislav D. Galkin
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  2. Elena V. Nozdrinova
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  3. Olga V. Pochinka
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Correspondence to Vladislav D. Galkin, Elena V. Nozdrinova or Olga V. Pochinka.

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Galkin, V.D., Nozdrinova, E.V. & Pochinka, O.V. Circular Fleitas Scheme for Gradient-Like Flows on the Surface. Regul. Chaot. Dyn. 28, 865–877 (2023). https://doi.org/10.1134/S1560354723060047

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