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Abelian integrals for a kind of non-Hamiltonian integrable systems under cubic polynomial perturbations. (English) Zbl 1215.34038

Summary: We investigate the number of isolated zeros of the Abelian integrals for a kind of non-Hamiltonian integrable systems with one center and two invariant straight lines and with other orbits formed by quartics. It is proved that the exact upper bound of the number of isolated zeros of the Abelian integrals under cubic polynomial perturbations is equal to two.

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
34A05 Explicit solutions, first integrals of ordinary differential equations
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