Yang, Jihua; Shang, Huahui On the number of zeros of abelian integrals for a kind of Hamiltonian systems of degree three. (Chinese. English summary) Zbl 1389.34104 J. Nat. Sci. Heilongjiang Univ. 34, No. 3, 291-296 (2017). Summary: By using the Picard-Fuchs equation method, the following perturbed Hamiltonian system \[ \begin{aligned}\dot{x} &=y+\varepsilon f(x, y), \\ \dot{y} &=-x-x^3+\varepsilon g(x,y)\end{aligned} \] is studied, where \(0<| \varepsilon| \ll 1\), \(f(x,y)\) and \(g(x,y)\) are polynomials of \(x\) and \(y\) of degrees \(n\). An upper bound \(B(n)\leq 3[\frac{n-1}{2}]\) is derived for the number of zeros of abelian integrals \(I| h |= \oint_{\Gamma_h}g(x,y)\mathrm{d}x-f(x,y)\mathrm{d}y\) on the open interval \((0,+\infty)\), where \(\Gamma_h\) is an oval lying on the algebraic curve \(H(x,y)=\frac{1}{2} y^2+\frac{1}{2} x^2+\frac{1}{4} x^4 = h\), \(h \in(0,+\infty)\). MSC: 34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) 34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 34E10 Perturbations, asymptotics of solutions to ordinary differential equations Keywords:abelian integrals; weakened Hilbert 16th problem; Picard-Fuchs equation; Hamiltonian system PDFBibTeX XMLCite \textit{J. Yang} and \textit{H. Shang}, J. Nat. Sci. Heilongjiang Univ. 34, No. 3, 291--296 (2017; Zbl 1389.34104) Full Text: DOI