Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

About History Editorial Page Addresses Scope Editorial Staff Submission Review For Authors Publication Ethics Issues Русская версия

About Boundedness of the Number of Compact Hypersurfaces of Foliations of Differential Systems

Author(s):

Valentine Yur'yevich Tyshchenko

Grodno state university,
Mathematics and computer science faculty

valentinet@mail.ru

Abstract:

Real singular foliations defined by Pfaff equations and autonomous systems of equations in total differentials are considered. Basing on the indexes of the lacunas of skew-symmetric tenzor fields, the Ostrogradsky formula and the degree of the map of vector fields criterions of boundedness of the number of compact invariant hypersurfaces of considered foliations have been obtained. The results have been adapted for the case two-dimensional autonomous system of ordinary differential equations. Illustrating examples are given.

Keywords

References:

  1. Problemy Gil’berta [Hilbert problems]. - Moscow: Nauka Publ., 1969. 240 p. (In Russian)
  2. Bendixon J. Sur les courbes defines par des equations differentielles. Acta Mathem, 1901, vol. 24, no. 1, pp. 1 - 88. (In French)
  3. Dulac H. Recherche des cycles limites. C. r. Acad. Sci, 1937, vol. 204, no. 23, pp. 1703 - 1706. (In French)
  4. Lloyd N. G. A note on the number of limit cycles in certain two-dimensional systems. London Math. Soc. , 1979, vol. 20, no 2, pp. 277 - 286
  5. Yamato K. An effective method of counting the number of limit cycles. Nagoya Math. J., 1979, vol. 79, pp. 35 - 114
  6. Gorbuzov V. N., Tyshchenko V. Yu. [Particular integrals of the systems of ordinary differential equatios]. Matematicheskii Sbornik, 1992, vol. 183, no. 3, pp. 76 - 94. (In Russian)
  7. Cherkas L. A. [Dulac function of polynomial autonomous systems on a plane]. Differentsial, nye Uravneniya, 1997, vol. 33, no. 5, pp. 689 - 699. (In Russian)
  8. Gorbuzov V. N. [Criterions of boundedness of number of possible compact hypersurfaces defined by differential systems]. Differentsial, nye Uravneniya, 1999, vol. 35, no. 1, pp. 30 - 37. (In Russian)
  9. Gorbuzov V. N. Compact integral manifolds of differentional systems. Mathematics. Dynamical Systems / arXiv:1009. 2998v1 [math. DS]. Cornell Univ., Ithaca, New York. New York , 2010. 27 p
  10. Tyshchenko V. Yu. [On compact invariant hypersurfaces of dyscrete dynamical systems]. Differentsial, nye Uravneniya, 2008, vol. 44, no. 7, pp. 1005 - 1006. (In Russian)
  11. Tyshchenko V. Yu. [Invariants of holomorphic multidimensional discrete dynamic systems ]. Differencial'nie uravnenia i processy upravlenia, 2013, no. 2 (In Russian). Available at: http://www. math. spbu. ru/diffjournal/pdf/tyshchenko2. pdfhttp://www. math. spbu. ru/diffjournal/pdf/tyshchenko2. pdf
  12. Rashevskii P. K. Geometricheskaya teoriya uravnenii s chastnymi proizvodnymi [The geometrical theory of the equations with partial derivatives]. Moscow, Leningrad: GITTL Publ., 1947. 355 p. (In Russian)
  13. Gorbuzov V. N. Integraly differencial'nih system [Integrals of differential systems]. Grodno: GrGU Publ., 2006. 447 p. (In Russian)
  14. Gorbuzov V. N. [Estimation of a number of compact layers of foliations defined by differential equations]. Differentsial, nye Uravneniya, 1997, vol. 33, no. 10, pp. 1307 - 1311. (In Russian)
  15. Ovsyannikov L. V. Gruppovoi analiz differencial'nih uravnenii [Group analysis of differential equations]. Moscow, Nauka Publ., 1978. 400 p. (In Russian)
  16. Dubrovin B. A., Novikov S. P., Fomenko A. T. Sovremennaya geometriya: Metody I prilozheniya [Modern geometry: Methods and applications]. Moscow: Nauka Publ., 1986. 760 p. (In Russian)
  17. Shilov G. E. Matematicheskii analiz (funkcii neskol’kikh peremennykh) [Mathematical analysis (functions of several variables]. Moscow: Nauka Publ., 1972. 623 p. (In Russian)
  18. Tkachev V. F., Tkachev Vl. F. [On criterions of absence of any and multiple limit cycles]. Matematicheskii Sbornik, 1960, vol. 52 (94), no. 3, pp. 811 - 822. (In Russian)
  19. Bogdanov Yu. S., Mazanik S. A., Syroid Yu. B. Kurs differencial'nih uravnenii [Course of differential equations]. Minsk: Universiteckae Publ, 1996. 287 p. (In Russian)

Full text (pdf)