The XVI-th Hilbert problem about limit cycles

Henryk Żołądek

Banach Center Publications (1995)

  • Volume: 34, Issue: 1, page 167-174
  • ISSN: 0137-6934

Abstract

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1. Introduction. The XVI-th Hilbert problem consists of two parts. The first part concerns the real algebraic geometry and asks about the topological properties of real algebraic curves and surfaces. The second part deals with polynomial planar vector fields and asks for the number and position of limit cycles. The progress in the solution of the first part of the problem is significant. The classification of algebraic curves in the projective plane was solved for degrees less than 8. Among general results we notice the inequalities of Harnack and Petrovski and Rohlin's theorem. Other results were obtained by Newton, Klein, Clebsch, Hilbert, Nikulin, Kharlamov, Gudkov, Arnold, Viro, Fidler. There are multidimensional generalizations: the theory of Khovansky, the inequalities of Petrovski and Oleinik, and others. In contrast to the algebraic part of the problem, the progress in the solution of the second part is small. In the present article we concentrate on the second part of the XVI-th Hilbert problem.

How to cite

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Żołądek, Henryk. "The XVI-th Hilbert problem about limit cycles." Banach Center Publications 34.1 (1995): 167-174. <http://eudml.org/doc/251301>.

@article{Żołądek1995,
abstract = {1. Introduction. The XVI-th Hilbert problem consists of two parts. The first part concerns the real algebraic geometry and asks about the topological properties of real algebraic curves and surfaces. The second part deals with polynomial planar vector fields and asks for the number and position of limit cycles. The progress in the solution of the first part of the problem is significant. The classification of algebraic curves in the projective plane was solved for degrees less than 8. Among general results we notice the inequalities of Harnack and Petrovski and Rohlin's theorem. Other results were obtained by Newton, Klein, Clebsch, Hilbert, Nikulin, Kharlamov, Gudkov, Arnold, Viro, Fidler. There are multidimensional generalizations: the theory of Khovansky, the inequalities of Petrovski and Oleinik, and others. In contrast to the algebraic part of the problem, the progress in the solution of the second part is small. In the present article we concentrate on the second part of the XVI-th Hilbert problem.},
author = {Żołądek, Henryk},
journal = {Banach Center Publications},
keywords = {Hilbert's 16th problem; number and position of the limit cycles; review; historical comments},
language = {eng},
number = {1},
pages = {167-174},
title = {The XVI-th Hilbert problem about limit cycles},
url = {http://eudml.org/doc/251301},
volume = {34},
year = {1995},
}

TY - JOUR
AU - Żołądek, Henryk
TI - The XVI-th Hilbert problem about limit cycles
JO - Banach Center Publications
PY - 1995
VL - 34
IS - 1
SP - 167
EP - 174
AB - 1. Introduction. The XVI-th Hilbert problem consists of two parts. The first part concerns the real algebraic geometry and asks about the topological properties of real algebraic curves and surfaces. The second part deals with polynomial planar vector fields and asks for the number and position of limit cycles. The progress in the solution of the first part of the problem is significant. The classification of algebraic curves in the projective plane was solved for degrees less than 8. Among general results we notice the inequalities of Harnack and Petrovski and Rohlin's theorem. Other results were obtained by Newton, Klein, Clebsch, Hilbert, Nikulin, Kharlamov, Gudkov, Arnold, Viro, Fidler. There are multidimensional generalizations: the theory of Khovansky, the inequalities of Petrovski and Oleinik, and others. In contrast to the algebraic part of the problem, the progress in the solution of the second part is small. In the present article we concentrate on the second part of the XVI-th Hilbert problem.
LA - eng
KW - Hilbert's 16th problem; number and position of the limit cycles; review; historical comments
UR - http://eudml.org/doc/251301
ER -

References

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  11. [11] E. Landis, I. Petrovski, On the number of limit cycles of the equation dy/dx=P(x,y)/Q(x,y), where P and Q are polynomials, Mat. Sb. 43 (1957), 149-168 (in Russian). 
  12. [12] G. S. Petrov, Complex zeroes of an elliptic integral, Functional Anal. Appl. 23 (1989), 160-161; Russian original: Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 88-89. 
  13. [13] A. Sommerfeld, Wellenmechanik, Frederic Ungar Publishing Co., New York (1937). 
  14. [14] Shi Songling, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. China Ser. A 11 (1979), 1051-1056 (in Chinese). 
  15. [15] A. N. Varchenko, Estimate of the number of zeroes of Abelian integrals depending on parameters and limit cycles, Functional Anal. Appl. 18 (1984), 98-108; Russian original: Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 14-25. Zbl0578.58035

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