Regular and Chaotic Dynamics

, Volume 21, Issue 2, pp 189–203 | Cite as

Efficient algorithms for the recognition of topologically conjugate gradient-like diffeomorhisms

  • Vyacheslav Z. GrinesEmail author
  • Dmitry S. Malyshev
  • Olga V. Pochinka
  • Svetlana Kh. Zinina


It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradient-like systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NPcomplete). We give polynomial-time algorithms for recognition of the corresponding graphs for two gradient-like systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, point-source potential-field models in efforts to determine the nature of the quiet-Sun magnetic field.


Morse–Smale diffeomorphism gradient-like diffeomorphism topological classification three-color graph directed graph graph isomorphism surface orientability surface genus polynomial-time algorithm magnetic field 

MSC2010 numbers

32S50 37C15 


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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Vyacheslav Z. Grines
    • 1
    Email author
  • Dmitry S. Malyshev
    • 1
    • 2
  • Olga V. Pochinka
    • 1
  • Svetlana Kh. Zinina
    • 3
  1. 1.National Research University Higher School of EconomicsNizhny NovgorodRussia
  2. 2.N. I. Lobachevsky State University of Nizhni NovgorodNizhny NovgorodRussia
  3. 3.Ogarev Mordovia State UniversitySaranskRussia

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