Nonlinear Oscillations

, Volume 9, Issue 2, pp 173–180 | Cite as

Transitive flows on orientable surfaces

  • O. A. Kadubovs’kyi


We consider smooth vector fields on closed orientable surfaces with a fixed collection of singularities and a finite number of separatrices none of which connects the equilibrium states. We prove that, on an orientable surface of arbitrary genus g ≥ 2, there exists a vector field with an admissible set of singularities (degenerate saddles) whose trajectory is everywhere dense on the surface.


Orientable Surface Morse Function Cyclic Order Periodic Trajectory Transitive Flow 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Kh. Aranson and V. Z. Grines, “Topological classification of flows on closed two-dimensional manifolds,” Usp. Mat. Nauk, 41, No. 1, 149–169 (1986).MathSciNetGoogle Scholar
  2. 2.
    S. Kh. Aranson and V. Z. Grines, “On some invariant dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for topological equivalence of transitive systems),” Mat. Sb., 90(132), No. 3, 372–402 (1973).MathSciNetGoogle Scholar
  3. 3.
    J. Palis, Jr., and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer, New York (1982).MATHGoogle Scholar
  4. 4.
    A. A. Blokhin, “Smooth ergodic flows on surfaces,” Tr. Mosk. Mat. Obshch., 27, 113–128 (1972).MATHGoogle Scholar
  5. 5.
    R. J. Sacker and J. R. Sell, “On the existence of periodic solutions on 2-manifolds,” J. Different. Equat., 11, No. 3, 449–463 (1972).CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    A. O. Prishlyak, “Vector fields with given collection of singularities,” Ukr. Mat. Zh., 49, No. 10, 1373–1384 (1997).CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    E. A. Girik, “On the existence of vector fields with given collection of singularities on a two-dimensional closed orientable manifold,” Ukr. Mat. Zh., 46, No. 12, 1706–1709 (1993).MathSciNetGoogle Scholar
  8. 8.
    D. N. Poltavets, “Cherry vector fields on the torus T 2,” Meth. Funct. Anal. Top., 2, No. 2, 94–98 (1996).MathSciNetMATHGoogle Scholar
  9. 9.
    A. A. Andronov, E. A. Leontovich, and A. G. Maier, Qualitative Theory of Dynamical Systems of the Second Order [in Russian], Nauka, Moscow (1966).Google Scholar
  10. 10.
    V. V. Sharko, “Smooth and topological equivalence of functions on surfaces,” Ukr. Mat. Zh., 55, No. 5, 687–700 (2003).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    O. A. Kadubovs’kyi, “Topological equivalence of functions on orientable surfaces,” Ukr. Mat. Zh., 58, No. 3, 343–351 (2006).Google Scholar
  12. 12.
    F. Takens, “The minimal number of critical points of a function on a compact manifold and the Lusternik-Schnirelman category,” Invent. Math., 6, 197–244 (1968).CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    A. O. Prishlyak, “Topological equivalence of smooth functions with isolated critical points on a closed surface,” Top. Appl., 119, 257–267 (2002).CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    M. Keane, “Interval exchange transformations,” Math. Z., 141, No. 5, 25–31 (1975).CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • O. A. Kadubovs’kyi
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKyiv

Personalised recommendations