Upper Bound on the Number of Zeros of Abelian Integrals for Hyperbolic Hamiltonian Systems of Degree Seven

Abstract

In this paper, we study the bound on the number of isolated zeros of Abelian integrals associated with the seventh-degree hyperbolic Hamiltonian system. When the unperturbed system has a unique period annulus bounded by a heteroclinic loop, a series of results on the upper bound of the number of zeros of the related Abelian integrals are obtained. However, there are still some open problems left about the exact bound. Furthermore, we give some results for the system when the parameters in the Hamiltonian system are fixed to certain values. The main tools involve algebraic symbolic computations such as counting the zeros of semi-algebraic systems.