The XVI-th Hilbert problem about limit cycles

• Autorzy: Henryk Żołądek
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1995
• Tom: 34
• Numer: 1
• Strony: 167-174

Daty

wydano

1995

Twórcy

autor

Henryk Żołądek

• Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

Bibliografia

• [1] N. N. Bautin, Du nombre de cycles limites naissant en cas de variation des coefficients d'un état d'équilibre du type foyer ou centre, Dokl. Akad. Nauk SSSR 24 (1939), 669-672.
• [2] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium point of focus or center type. Amer. Math. Soc. Transl. Ser. 1, 5 (1962), 396-419. (Russian original: Mat. Sb. 30 (1952), 181-196).
• [3] I. Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 25 (1901), 1-88.
• [4] H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923).
• [5] J. Ecalle, J. Martinet, R. Moussu, J. P. Ramis, Non-accumulation de cycles limites, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 375-378, 431-434.
• [6] J. Ecalle, Finitude des cycles limites et accelero-sommation de l'application de ratour, in: Bifurcations of Planar Vector Fields, Proc. Luminy 1989, Lecture Notes in Math. 1455, Springer-Verlag (1990), 74-159.
• [7] D. Hilbert, Mathematische Probleme, II ICM Paris 1900, Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. (1900) 253-297.
• [8] Yu. S. Il'iashenko, Finiteness theorems for limit cycles, Uspekhi Mat. Nauk 45 (1991), 143-200 (in Russian).
• [9] Yu. S. Il'iashenko, Finiteness theorems for limit cycles, AMS Monographs (1991).
• [10] 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 of the second degree, Mat. Sb. 37 (1955), 209-250 (in Russian).
• [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] 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] A. Sommerfeld, Wellenmechanik, Frederic Ungar Publishing Co., New York (1937).
• [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] 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.