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The number of zeros of abelian integrals for a perturbation of hyperelliptic Hamiltonian system with degenerated polycycle. (English) Zbl 1270.34057

Summary: We provide a complete study of the zeros of abelian integrals obtained by integrating the 1-form \((\alpha + \beta x + x^2)ydx\) over the compact level curves of the hyperelliptic Hamiltonian \(H(x,y)=\frac{y^{2}}{2}-\frac{1}{5}x^3(x-1)^2\). Such a family of compact level curves is bounded by a polycycle passing through a nilpotent cusp and a hyperbolic saddle of this hyperelliptic Hamiltonian system, which is not the exceptional family of ovals proposed by Gavrilov and Iliev. It is shown that the least upper bound for the number of zeros of the related hyperelliptic abelian integral is two, and this least upper bound can be achieved for some values of parameters ({\(\alpha\)}, {\(\beta\)}). This implies that the abelian integral still has Chebyshev property for this nonexceptional family of ovals. Moreover, we derive the asymptotic expansion of abelian integrals near a polycycle passing through a nilpotent cusp and a hyperbolic saddle in a general case.

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
34C37 Homoclinic and heteroclinic solutions to 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
34E05 Asymptotic expansions of solutions to ordinary differential equations
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