Zhang, Erli; Xing, Yuqing Limit cycle bifurcations for a kind of Hamilton systems of degree three. (Chinese. English summary) Zbl 1399.37033 Acta Math. Sci., Ser. A, Chin. Ed. 37, No. 5, 825-833 (2017). Summary: By using the Picard-Fuchs equation method, we obtain an upper bound of the number of zeros of abelian integrals \(I\left(h \right)={\oint_{{\Gamma_h}}}g\left({x, y} \right)dx-f\left({x, y} \right)dy\), where \({\Gamma_h}\) is the closed orbit defined by \(H\left({x, y} \right)={x^2}+{y^2}+2xy+\frac{1}{a} (x^4+y^4)=h\), \(a > 0\), \(h \in (0, +\infty)\), \(f (x, y)\) and \(g (x, y)\) are real polynomials in \(x\) and \(y\) of degree \(n\). Therefore, we get the upper bound of the number of limit cycles of this system. MSC: 37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations Keywords:Hamilton system; abelian integral; Picard-Fuchs equation; limit cycle PDFBibTeX XMLCite \textit{E. Zhang} and \textit{Y. Xing}, Acta Math. Sci., Ser. A, Chin. Ed. 37, No. 5, 825--833 (2017; Zbl 1399.37033)