Yang, Jihua; Sui, Shiyou; Zhao, Liqin On the number of zeros of abelian integral for a class of cubic Hamilton systems with the phase portrait “butterfly”. (English) Zbl 1431.34043 Qual. Theory Dyn. Syst. 18, No. 3, 947-967 (2019). Summary: The present paper is devoted to study the number of zeros of abelian integral for the near-Hamilton system \[\begin{cases} \dot{x} = 2y(bx^2 + 2cy^2) + \varepsilon f(x,y),\\ \dot{y} = 2x(1 - 2ax^2 - by^2) + \varepsilon g(x,y),\end{cases}\] where \(a,b,c \in \mathbb{R}, b < 0, c > 0, b^2 < 4ac, 0 < |\varepsilon| \ll 1, f(x, y)\) and \(g(x, y)\) are polynomials in \((x, y)\) of degree \(n\). The generators of the corresponding abelian integral satisfy three different Picard-Fuchs equations. We obtain an upper bound of the number of isolated zeros of the abelian integral. Cited in 2 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 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations Keywords:Hamilton system; abelian integral; weakened Hilbert’s 16th problem; Picard-Fuchs equation; Chebyshev space PDFBibTeX XMLCite \textit{J. Yang} et al., Qual. Theory Dyn. Syst. 18, No. 3, 947--967 (2019; Zbl 1431.34043) Full Text: DOI References: [1] Arnold, V.: Ten problems. In: Theory of Singularities and Its Applications. 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