Qi, Minghui; Zhao, Liqin Bifurcations of limit cycles from a quintic Hamiltonian system with a figure double-fish. (English) Zbl 1275.34049 Int. J. Bifurcation Chaos Appl. Sci. Eng. 23, No. 7, Article ID 1350116, 15 p. (2013). Summary: In this paper, we consider Liénard systems of the form \[ \frac{dx}{dt}=y, \quad \frac{dy}{dt}=-\bigg(-x+\frac{21}{10}x^3\bigg)+\epsilon(\alpha+\beta x^2+\gamma x^4)y, \] where \(0 < |\epsilon| \ll 1\) and \((\alpha, \beta, \gamma) \in \mathbb R^3\). We prove that the least upper bound of the number of isolated zeros of the related abelian integrals \[ I(h)=\oint_{\Gamma_{h}}(\alpha+\beta x^2+\gamma x^4)ydx \] is 4 (counting the multiplicity) and this upper bound is a sharp one. Cited in 3 Documents 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 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations Keywords:hyper-elliptic Hamiltonian system; abelian integral; period annulus; Picard-Fuchs equation PDFBibTeX XMLCite \textit{M. Qi} and \textit{L. Zhao}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 23, No. 7, Article ID 1350116, 15 p. (2013; Zbl 1275.34049) Full Text: DOI References: [1] DOI: 10.1016/j.na.2007.02.039 · Zbl 1158.34024 [2] DOI: 10.1016/j.na.2007.10.054 · Zbl 1161.34329 [3] DOI: 10.1016/j.camwa.2009.12.024 · Zbl 1189.34072 [4] DOI: 10.1016/0022-0396(87)90122-7 · Zbl 0622.34033 [5] DOI: 10.1006/jdeq.2000.3977 · Zbl 1004.34018 [6] DOI: 10.1006/jdeq.2000.3978 · Zbl 1034.34036 [7] DOI: 10.1016/S0022-0396(02)00110-9 · Zbl 1056.34044 [8] DOI: 10.1016/S0022-0396(02)00111-0 · Zbl 1057.34015 [9] DOI: 10.1007/s10884-008-9108-3 · Zbl 1165.34016 [10] DOI: 10.1112/plms/s3-69.1.198 · Zbl 0802.58046 [11] DOI: 10.1006/jdeq.1996.0017 · Zbl 0849.34022 [12] DOI: 10.1088/0951-7715/15/6/310 · Zbl 1219.34042 [13] Pontryagin L., Zh. Exp. Th. Phys. 4 pp 234– (1934) [14] DOI: 10.1016/j.jde.2010.11.004 · Zbl 1217.34051 [15] DOI: 10.1016/j.physleta.2006.05.031 · Zbl 1142.34327 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.