Yang, Jihua; Zhang, Erli; Li, Sanjie; Liu, Mei The number of zeros of abelian integrals for near-Hamilton system of degree three with a butterfly. (Chinese. English summary) Zbl 1413.34137 Acta Math. Sin., Chin. Ser. 61, No. 1, 19-26 (2018). Summary: We consider the following near-Hamilton system \[ \begin{aligned} \dot{x} &= 2y\left ({a{x^2} + 2c{y^2}} \right) + \varepsilon f\left ({x, y} \right), \\ \dot{y} &= 2x\left ({1 - 2b{x^2} - a{y^2}} \right) + \varepsilon g\left ({x, y} \right), \end{aligned} \] where \(a < 0\), \(c > 0\), \(4bc > {a^2}\), \(0 < \left| \varepsilon \right| \ll 1\), \(f\left ({x, y} \right)\) and \(g\left ({x, y} \right)\) are cubic polynomials of \(x\) and \(y\). We obtain the upper bound of the number of isolated zeros of the abelian integral. 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 Keywords:Hamilton system; abelian integral; Picard-Fuchs equation; butterfly PDFBibTeX XMLCite \textit{J. Yang} et al., Acta Math. Sin., Chin. Ser. 61, No. 1, 19--26 (2018; Zbl 1413.34137)