On Algorithms that Effectively Distinguish Gradient-Like Dynamics on Surfaces

  • Vladislav E. KruglovEmail author
  • Dmitry S. Malyshev
  • Olga V. Pochinka
Research Contribution


In the present paper we survey existing graph invariants for gradient-like flows on surfaces up to the topological equivalence and develop effective algorithms for their distinction (let us recall that a flow given on a surface is called a gradient-like flow if its non-wandering set consists of a finite set of hyperbolic fixed points, and there is no trajectories connecting saddle points). Additionally, we construct a parametrized algorithm for the Fleitas’s invariant, which will be of linear time, when the number of sources is fixed. Finally, we prove that the classes of topological equivalence and topological conjugacy are coincide for gradient-like flows, so, all the proposed invariants and distinguishing algorithms works also for topological classification, taking in sense time of moving along trajectories. So, as the main result of this paper we have got multiple ways to recognize equivalence and conjugacy class of arbitrary gradient-like flow on a closed surface in a polynomial time.


Gradient-like dynamics Effective distinguishing algorithms Graph invariants Surface 



The algorithm for the Peixoto invariants was created with a support of the Russian Foundation for Basic Research Project 18-31-00022 mol\(\underline{\,\,\,}\)a. The polynomial-time algorithm for the Wang invariant was obtained as an output of the research project of the Basic Research Program at the National Research University Higher School of Economics (HSE) in 2019. The algorithms for the Fleitas’ graph invariant was created under a support of RF President Grant MD-879.2019.1 and Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of Economics. The polynomial-time algorithm for the Oshemkov-Sharko invariant and conjugacy theorem were implemented in the framework of the Russian Science Foundation Project 17-11-01041.


  1. Andronov, A.A., Pontryagin, L.S.: Rough systems. Doklady Akademii Nauk SSSR 14(5), 247–250 (1937). (in Russian) Google Scholar
  2. Aho, A.V., Corasick, M.J.: Efficient string matching: An aid to bibliographic search. Commun ACM. 18(6), 333–340 (1975)Google Scholar
  3. Apostolico, A., Giancarlo, R.: The Boyer–Moore–Galil string searching strategies revisited. SIAM J Comput 15(1), 98–105 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cobham, A.: The intrinsic computational difficulty of functions. In: Proc 1964 International Congress for Logic, Methodology, and Philosophy of Science, pp. 24–30. North-Holland, Amsterdam (1964)Google Scholar
  5. Donald, K., Morris, J.H., Vaughan, P.: Fast pattern matching in strings. SIAM J Comput 6(2), 323–350 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Fleitas, G.: Classification of gradiet-like flows on dimensions two and three. Bol. Soc. Bras. Mat. 6, 155–183 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Galil, Z., Hoffmann, C., Schnorr, C., Weber, A.: An \(O(n^3log n)\) deterministic and an \(O(n^3)\) Las Vegas isomorphism test for trivalent graphs. J. ACM 34, 513–531 (1987)CrossRefGoogle Scholar
  8. Grines, V., Medvedev, T., Pochinka, O.: Dynamical Systems on 2- and 3-Manifolds. Springer, Basel (2016)CrossRefzbMATHGoogle Scholar
  9. Hopcroft, J.E., Wong, J.K.: Linear time algorithm for isomorphism of planar graphs: preliminary report. In: Proceedings of the 6th Annual ACM Symposium on Theory of Computing, Seattle, Wash., pp. 172–184 (1974)Google Scholar
  10. Kawarabayashi, K.: Graph isomorphism for bounded genus graphs in linear time (2015). arXiv:1511.02460
  11. Kruglov, V.: Topological conjugacy of gradient-like flows on surfaces. Dinamicheskie sistemy 8(36)(1), 15–21 (2018)MathSciNetzbMATHGoogle Scholar
  12. Kruglov, V.E., Malyshev, D.S., Pochinka, O.V.: A multicolour graph as a complete topological invariant for \(\Omega \)-stable flows without periodic trajectories on surfaces. Sb. Math. 209(1), 96–121 (2018a). MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kruglov, V.E., Malyshev, D.S., Pochinka, O.V.: Topological classification of \(\Omega \)-stable flows on surfaces by means of effectively distinguishable multigraphs. Discret. Contin. Dyn. Syst. 38(9), 4305–4327 (2018b). MathSciNetCrossRefzbMATHGoogle Scholar
  14. Leontovich, E.A., Mayer, A.G.: About trajectories determining qualitative structure of sphere partition into trajectories. Doklady Akademii Nauk SSSR 14(5), 251–257 (1937). (in Russian) Google Scholar
  15. Leontovich, E.A., Mayer, A.G.: About scheme determining topological structure of partition into trajectories. Doklady Akademii Nauk SSSR 103(4), 557–560 (1955). (in Russian) MathSciNetGoogle Scholar
  16. Luks, E.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25, 42–65 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Miller, G.: Isomorphism testing for graphs of bounded genus. In: Proceeding of the 12th Annual ACM Symposium on Theory of Computing, pp. 225–235 (1980)Google Scholar
  18. Oshemkov, A.A., Sharko, V.V.: Classification of Morse–Smale flows on two-dimensional manifolds. Sb. Math. 189(8), 1205–1250 (1998). MathSciNetCrossRefzbMATHGoogle Scholar
  19. Palis, J., De Melo, W.: Geometric Theory of Dynamical Systems: An Introduction. Translation from the Portuguese by Manning AK. Springer, New York (1982)CrossRefzbMATHGoogle Scholar
  20. Peixoto, M.M.: On the classification of flows on 2-manifolds. In: Dynamical systems Proceedings of Symposium Held at the University of Bahia, Salvador, Brazil, pp. 389–419 (1973)Google Scholar
  21. Pesin, YaB, Yurchenko, A.A.: Some physical models of the reaction–diffusion equation, and coupled map lattices. Russ. Math. Surv. 59(3), 481–513 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  22. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  23. Wang, X.: The \(C^*\)-algebras of Morse–Smale flows on two-manifolds. Ergod. Theory Dyn. Syst. 10, 565–597 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2019

Authors and Affiliations

  1. 1.HSENizhny NovgorodRussia

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