Song, Yan Abelian integrals for a kind of non-Hamiltonian integrable systems under cubic polynomial perturbations. (English) Zbl 1215.34038 Int. J. Bifurcation Chaos Appl. Sci. Eng. 21, No. 3, 789-798 (2011). Summary: We investigate the number of isolated zeros of the Abelian integrals for a kind of non-Hamiltonian integrable systems with one center and two invariant straight lines and with other orbits formed by quartics. It is proved that the exact upper bound of the number of isolated zeros of the Abelian integrals under cubic polynomial perturbations is equal to two. MSC: 34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) 34E10 Perturbations, asymptotics of solutions to ordinary differential equations 34A05 Explicit solutions, first integrals of ordinary differential equations Keywords:non-Hamiltonian integrable system; Picard-Fuchs equation; cubic polynomial perturbations; Abelian integral PDFBibTeX XMLCite \textit{Y. Song}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 21, No. 3, 789--798 (2011; Zbl 1215.34038) Full Text: DOI References: [1] Chen G., Discr. Contin. Dyn. Syst. 16 pp 157– [2] DOI: 10.1006/jdeq.1997.3285 · Zbl 0883.34035 [3] DOI: 10.1006/jdeq.2000.3977 · Zbl 1004.34018 [4] Dumortier F., J. Diff. Eqs. 176 pp 209– [5] DOI: 10.1016/S0022-0396(02)00110-9 · Zbl 1056.34044 [6] DOI: 10.1016/S0022-0396(02)00111-0 · Zbl 1057.34015 [7] Hartman P., Ordinary Differential Equations (1982) · Zbl 0476.34002 [8] Horozov E., Proc. London Math. Soc. 69 pp 198– [9] DOI: 10.1016/S0007-4497(98)80080-8 · Zbl 0920.34037 [10] DOI: 10.1088/0951-7715/18/1/016 · Zbl 1077.34035 [11] DOI: 10.1006/jdeq.1996.0017 · Zbl 0849.34022 [12] Li W. G., Normal Form Theory and Their Application (2000) [13] DOI: 10.1088/0951-7715/15/3/321 · Zbl 1038.37016 [14] DOI: 10.1016/S0022-247X(02)00018-5 · Zbl 1019.34042 [15] Zhao Y. L., Discr. Contin. Dyn. Syst. 3 pp 795– [16] DOI: 10.1006/jdeq.1994.1049 · Zbl 0797.34044 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.