Zhou, Xin; Li, Cuiping On the algebraic structure of abelian integrals for a kind of perturbed cubic Hamiltonian systems. (English) Zbl 1180.34033 J. Math. Anal. Appl. 359, No. 1, 209-215 (2009). The authors study the algebraic structure of abelian integrals for a class of perturbed cubic Hamiltonian systems, which is related to the so-called weakened Hilbert’s 16th problem. Reviewer: Valery A. Gaiko (Minsk) Cited in 8 Documents MSC: 34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations 34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) Keywords:cubic Hamiltonian system; abelian integral; Picard-Fuchs equation; weakened Hilbert’s 16th problem PDFBibTeX XMLCite \textit{X. Zhou} and \textit{C. Li}, J. Math. Anal. Appl. 359, No. 1, 209--215 (2009; Zbl 1180.34033) Full Text: DOI References: [1] Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations (1988), Springer-Verlag: Springer-Verlag New York [2] Arnold, V. I., Ten problems, Adv. Soviet Math., 1, 1-8 (1990) [3] Khovansky, A. G., Real analytic manifolds with finiteness properties and complex Abelian integrals, Funct. Anal. Appl., 18, 119-128 (1984) [4] Varchenko, A. N., Estimate of the number of zeros of an Abelian integral depending on a parameter and limit cycles, Funct. Anal. Appl., 18, 90-108 (1984) · Zbl 0578.58035 [5] Horozov, E.; Iliev, I. D., On the number of limit cycles in perturbations of quadratic Hamiltonian systems, Proc. London Math. Soc., 69, 198-224 (1994) · Zbl 0802.58046 [6] Gavrilov, L., The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math., 143, 449-497 (2001) · Zbl 0979.34024 [7] Markov, Y., Limit cycles of perturbations of a class of quadratic Hamiltonian vector fields, Serdica Math. J., 22, 91-108 (1996) · Zbl 0859.58023 [8] Li, C.; Zhang, Z., Remarks on 16th weak Hilbert problem for \(n = 2\), Nonlinearity, 15, 1975-1992 (2002) · Zbl 1219.34042 [9] Chen, F.; Li, C.; Llibre, J., A unified proof on the weak Hilbert 16th problem for \(n = 2\), J. Differential Equations, 221, 309-342 (2006) · Zbl 1098.34024 [10] Horozov, E.; Iliev, I. D., Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians, Nonlinearity, 11, 1521-1537 (1998) · Zbl 0921.58044 [11] Zhao, Y.; Zhang, Z., Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians, J. Differential Equations, 155, 73-88 (1999) · Zbl 0961.34016 [12] Petrov, G. S., Number of zeros of complete elliptic integrals, Funct. Anal. Appl., 18, 148-149 (1984) · Zbl 0567.14006 [13] Petrov, G. S., Elliptic integrals and their nonoscillation, Funct. Anal. Appl., 20, 37-40 (1986) · Zbl 0656.34017 [14] Petrov, G. S., Complex zeros of an elliptic integral, Funct. Anal. Appl., 21, 247-248 (1987) · Zbl 0633.33002 [15] Petrov, G. S., Complex zeros of an elliptic integral, Funct. Anal. Appl., 23, 160-161 (1989) · Zbl 0709.33015 [16] Rousseau, C.; Zoladek, H., Zeroes of complete elliptic integrals for 1:2 resonance, J. Differential Equations, 94, 41-54 (1991) · Zbl 0738.33014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.