Mathematical Problems
(lecture derived on the International Congress of Mathematicians, París, 1900)
D. Hilbert,
Bulletin of the AMS, 37, 4, pp. 407-436 (Reprinted from Bull. Amer.
Math. Soc. 8, July 1902, pp. 437–479. Originally published as
Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematike-Congresszu,
Paris 1900, Gött. Nachr.1900, 253-297, Vandenhoeck & Ruprecht, Göttingen.
Translated for the Bulletin, with the author’s permission, by Dr. Mary Winston,
Newson,1902.)
Centennial history of Hilbert's 16th problem
Y. Ilyashenko
Bull. (New series) of the A.M.S., 39, 3, 2002, pp. 301-354.
A
Sideways Look at Hilbert’s Twenty-three Problems of 1900
Ivor Grattan-Guinness
Notices of the A.M.S., 47, 7, pp. 757-757.
(5 ciclos límite) On certain
generalization of Bautin's theorem
H. Zolakek
Nonlinearity, 7, 1994, pp. 273-279.
Variational approach to a class of nonlinear oscillators with several limit
cycles
M. C. Depassier, J. Mura
Physical Review E, 64, 2001, pp. 056217-1 a 056217-6
(23 ciclos límite)
Investigations of bifurcations of limit cycles in Z2-equivalent planar vector
fields of degree 5
J. Li, H. S. Y. Chan, K. W. Chung
International Journal of Bifurcation and Chaos, 12, 10, 2002, pp.
2137-2157.
(14 ciclos límite) Fourteen
limit cycles in a cubic Hamiltonian system with nine-order perturbed term
M. Tang, X. Hong
Chaos, Solitons and Fractals, 14, 2002, pp.1361–1369.
On the
Limit Cycles of Quadratic Differential Systems
X. Zhang
Acta Mathematica Sinica, English Series, 18, 4, 2002, pp. 803–816.
Abelian
Integrals and Limit Cycles in Polynomial Dynamical Systems
Valery A. Gaiko
Nonlinear Phenomena in Complex Systems, 6, 1, 2003, pp. 577-581.
Number
and amplitude of limit cycles emerging from topologically equivalent perturbed
centers
J. L. López, R. López-Ruiz
Chaos, Solitons and Fractals, 17, 2003, pp. 135–143.
(5 ciclos límite) The
same distribution of limit cycles in five perturbed cubic Hamiltonian systems
Z.Liu, Z. Yang, T. Jiang
International Journal of Bifurcation and Chaos, 13, 1, 2003, pp.
243-249.
On the second part of Hilbert’s 16th problem
E. Oxenhielm,
Nonlinear Analysis, Received 3 July 2003; accepted 3 October 2003.
(de Wikipedia) Elin Oxenhielm es
una estudiante de matemáticas sueca que en diciembre de 2003 afirmó erróneamente
haber resuelto el problema 16 de Hilbert. Su artículo había sido aceptado en la
revista Nonlinear Analysis, pero después de otras revisiones la aceptación fue
rechazada. Un anuncio adicional del Editor en Jefe V.
Lakshmikantham dice: "This paper has been withdrawn from the Articles in Press
section of ScienceDirect. Further referee reports, which provide specific
details, show that the proof offered does not stand on a rigorous foundation and
that further work is necessary." Más información:
(10 ciclos límite) A study
on the existence of limitcycles of a planar system with third-degree polynomials
M. Han, Y. Lin, P. Yu
International Journal of Bifurcation and Chaos, 14, 1, 2004, pp.
41-60.
(11 ciclos límite) On
the number and distribution of limit cycles in a cubic system
M. Han, T. Zhang, H. Zang.
International Journal of Bifurcation and Chaos, 14, 12, 2004, pp.
4285-4292.
(23 ciclos límite) On the
control of parameters of distributions of limit cycles for a Z2-equivariant
perturbed planar hamitonian polynomial vectro field
J. Li, H. Zhou
International Journal of Bifurcation and Chaos, 15, 1, 2005, pp. 137–155.
(12 ciclos límite) Twelve
limit cycles in a cubic case of the 16th Hilbert problem
P. Yu, M. Han
International Journal of Bifurcation and Chaos, 15, 7, 2005 pp.
2191–2205.
(35 ciclos límite)
Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order
perturbation
S. Wang, P. Yu
Chaos, Solitons and Fractals, 26, 2005, pp. 1317–1335.
(11 ciclos límite) The
number and distributions of limit cycles for a class of cubic near-Hamiltonian
systems
H. Zang, T. Zhang, M. Han
J. Math. Anal. Appl., 316, 2006, pp. 679–696.
(50 ciclos límite)
Bifurcations of limit cycles in a Z2-equivariant planar polynomial vector field
of degree 7
J. Li, M. Zhang, S. Li
International Journal of Bifurcation and Chaos, 16, 4, 2006,
pp. 925–943
(2 ciclos límite) A unified
proof on the weak Hilbert 16th problem for n = 2
F. Chena, C. Lia, J. Llibreb, Z. Zhanga
J. Differential Equations, 221, 2006, pp. 309 – 342.
(13 ciclos límite) Thirteen
limit cycles for a class of Hamiltonian systems under seven-order perturbed
terms
G. Tigan
Chaos, Solitons and Fractals, 31, 2007, pp. 480–488.
(80 ciclos límite)
Bifurcation of Limit Cycles in Z-equivariant vector fields of degree 9
S. Wang, P. Yu, J. Li
International Journal of Bifurcation and Chaos, 16, 8, 2006, pp.
2309–2324.
(n(n+1)/2-1 ciclos límite)
On the Number Of Limit Cycles in Near-Hamiltonian Polynomial Systems
M. Han, G. Chen, C. J. Sun,
International Journal of Bifurcation and Chaos, 17, 6, 2007, pp.
2033–2047.
En un sistema Hamiltoniano con perturbaciones polinomiales de orden n
puede haber a lo sumo n(n+1)/2-1
ciclos límites alrededor de un centro.
