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The number of zeros of abelian integrals for a kind of quartic Hamiltonians with a nilpotent center. (Chinese. English summary) Zbl 1374.34095

Summary: In this paper, we prove that the number of zeros of the abelian integral for the Hamiltonians \[ H(x,y) = -x^2+ ax^2y^2 + bx^4 + cy^4 \] on the interval \((0,\frac{c}{a^2-4bc})\) is not more than \(3n+3[\frac{n-1}{4}]+14\) (taking into account the multiplicity), where \(a > 0, b < -2, c < 0, a^2 > 4bc\).

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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