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On the number of zeros of abelian integral for a class of cubic Hamilton systems with the phase portrait “butterfly”. (English) Zbl 1431.34043

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.

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
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