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Mat. Sb., 1998, Volume 189, Number 8, Pages 93–140 (Mi msb341)  

This article is cited in 23 scientific papers (total in 23 papers)

Classification of Morse–Smale flows on two-dimensional manifolds

A. A. Oshemkova, V. V. Sharkob

a M. V. Lomonosov Moscow State University
b Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: The problem of topological trajectory classification of Morse–Smale flows on closed two-dimensional surfaces is considered. Important results in this direction have been obtained by Peixoto and his school. However, the complete solution of this problem has not yet been accurately presented. The new topological invariants constructed in our work have a simpler form than those in the works of Peixoto. In particular, a list of Morse–Smale flows of small complexity is given which has been obtained by the authors by means of the invariants constructed by them.


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English version:
Sbornik: Mathematics, 1998, 189:8, 1205–1250

Bibliographic databases:

UDC: 513.83
MSC: 34C35, 58F09
Received: 13.11.1997

Citation: A. A. Oshemkov, V. V. Sharko, “Classification of Morse–Smale flows on two-dimensional manifolds”, Mat. Sb., 189:8 (1998), 93–140; Sb. Math., 189:8 (1998), 1205–1250

Citation in format AMSBIB
\by A.~A.~Oshemkov, V.~V.~Sharko
\paper Classification of Morse--Smale flows on two-dimensional manifolds
\jour Mat. Sb.
\yr 1998
\vol 189
\issue 8
\pages 93--140
\jour Sb. Math.
\yr 1998
\vol 189
\issue 8
\pages 1205--1250

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    This publication is cited in the following articles:
    1. Plachta L., “The combinatorics of gradient-like flows and foliations on closed surfaces. I. Topological classification”, Topology Appl., 128:1 (2003), 63–91  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    2. A. O. Prishlyak, “Complete topological invariants of Morse–Smale flows and handle decompositions of 3-manifolds”, J. Math. Sci., 144:5 (2007), 4492–4499  mathnet  crossref  mathscinet  zmath  elib
    3. E. V. Zhuzhoma, V. S. Medvedev, “Global Dynamics of Morse–Smale Systems”, Proc. Steklov Inst. Math., 261 (2008), 112–135  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. E. A. Kudryavtseva, I. M. Nikonov, A. T. Fomenko, “Maximally symmetric cell decompositions of surfaces and their coverings”, Sb. Math., 199:9 (2008), 1263–1353  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. O. V. Pochinka, “Neobkhodimye i dostatochnye usloviya topologicheskoi sopryazhennosti kaskadov Morsa–Smeila na 3-mnogoobraziyakh”, Nelineinaya dinam., 7:2 (2011), 227–238  mathnet  elib
    6. da Silva A.R., “Peixoto Classification of 2-Dim Flows Revisited”, Dynamics, Games and Science II, Springer Proceedings in Mathematics, 2, eds. Peixoto M., Pinto A., Rand D., Springer-Verlag Berlin, 2011, 639–645  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    7. V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, O. V. Pochinka, “On embedding a Morse-Smale diffeomorphism on a 3-manifold in a topological flow”, Sb. Math., 203:12 (2012), 1761–1784  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. V. Z. Grines, O. V. Pochinka, “Morse–Smale cascades on 3-manifolds”, Russian Math. Surveys, 68:1 (2013), 117–173  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. V. Z. Grines, E. Ya. Gurevich, O. V. Pochinka, “The Energy Function of Gradient-Like Flows and the Topological Classification Problem”, Math. Notes, 96:6 (2014), 921–927  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. V. Z. Grines, S. H. Kapkaeva, O. V. Pochinka, “A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces”, Sb. Math., 205:10 (2014), 1387–1412  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Vyacheslav Z. Grines, Dmitry S. Malyshev, Olga V. Pochinka, Svetlana Kh. Zinina, “Efficient Algorithms for the Recognition of Topologically Conjugate Gradient-like Diffeomorhisms”, Regul. Chaotic Dyn., 21:2 (2016), 189–203  mathnet  crossref  mathscinet
    12. V. Z. Grines, E. V. Zhuzhoma, O. V. Pochinka, “Sistemy Morsa–Smeila i topologicheskaya struktura nesuschikh mnogoobrazii”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 61, RUDN, M., 2016, 5–40  mathnet
    13. Martinez-Alfaro J., Meza-Sarmiento I.S., Oliveira R.D.S., “Singular levels and topological invariants of Morse Bott integrable systems on surfaces”, J. Differ. Equ., 260:1 (2016), 688–707  crossref  mathscinet  zmath  isi  scopus
    14. Maksymenko S. Polulyakh E., “Foliations With All Non-Closed Leaves on Non-Compact Surfaces”, Methods Funct. Anal. Topol., 22:3 (2016), 266–282  mathscinet  zmath  isi
    15. V. E. Kruglov, D. S. Malyshev, O. V. Pochinka, “Grafovyi kriterii topologicheskoi ekvivalentnosti $\Omega$-ustoichivykh potokov bez periodicheskikh traektorii na poverkhnostyakh i effektivnyi algoritm dlya ego primeneniya”, Zhurnal SVMO, 18:2 (2016), 47–58  mathnet  elib
    16. V. E. Kruglov, O. V. Pochinka, “Grafovyi kriterii topologicheskoi ekvivalentnosti $\Omega$-ustoichivykh potokov na poverkhnostyakh”, Zhurnal SVMO, 18:3 (2016), 41–48  mathnet  elib
    17. V. Z. Grines, O. V. Pochinka, “Postroenie energeticheskikh funktsii dlya $\Omega$-ustoichivykh diffeomorfizmov na $2$- i $3$-mnogoobraziyakh”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 63, no. 2, Rossiiskii universitet druzhby narodov, M., 2017, 191–222  mathnet  crossref
    18. V. E. Kruglov, G. N. Talanova, “O poverkhnostyakh, skleennykh iz 2n-ugolnikov”, Zhurnal SVMO, 19:3 (2017), 31–40  mathnet  crossref  elib
    19. V. E. Kruglov, D. S. Malyshev, O. V. Pochinka, “A multicolour graph as a complete topological invariant for $\Omega$-stable flows without periodic trajectories on surfaces”, Sb. Math., 209:1 (2018), 96–121  mathnet  crossref  crossref  adsnasa  isi  elib
    20. Martinez-Alfaro J., Meza-Sarmiento I.S., Oliveira R.D.S., “Singular Levels and Topological Invariants of Morse-Bott Foliations on Non-Orientable Surfaces”, Topol. Methods Nonlinear Anal., 51:1 (2018), 183–213  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    21. Kruglov V. Malyshev D. Pochinka O., “Topological Classification of Omega-Stable Flows on Surfaces By Means of Effectively Distinguishable Multigraphs”, Discret. Contin. Dyn. Syst., 38:9 (2018), 4305–4327  crossref  mathscinet  zmath  isi  scopus
    22. V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, O. V. Pochinka, “Classification of Morse–Smale systems and topological structure of the underlying manifolds”, Russian Math. Surveys, 74:1 (2019), 37–110  mathnet  crossref  crossref  adsnasa  isi  elib
    23. V. Z. Grines, E. Ya. Gurevich, E. D. Kurenkov, “Topological Classification of Gradient-Like Flows with Surface Dynamics on $3$-Manifolds”, Math. Notes, 107:1 (2020), 173–176  mathnet  crossref  crossref  isi
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