An Estimate Of The Number Of Limit Cycles Of A One-parameter Hamiltonian System Under Polynomial Perturbations

It's studied that the number of isolated zeros of the Mel'nikov function for a one-parameter Hamiltonian system under polynomial perturbations in this paper. The per-turbed system is (?)= -2y + 3λy² + (?)P(x, y), (?)= -x - x² + (?)Q(x, y), whereλis a small parameter, (?) is a perturbed parameter, (?),k is a positive integer big enough, degP, degQ≤n, and n is a non-negative integer. The system is simplified to a Bogdonov-Takens system with one-parameter by using coordinate transformation, which is (?) = y - 3λy² + (?)P^*(x, y), (?)= -x + x² + (?)Q^*(x, y), where degP^*, degQ^*≤n. According to the first-order Mel'nikov function M₁(h,λ) in terms of the Taylor expansion about the small parameter A, one upper bound of the number of isolated zeros of M₁ (h,λ) is given by Petrov's theorem. When (?)m/(?)λM₁(h,λ)|_λ=0 (?)0(m is a non-negative integer), B(2, n)≤n + m-1. Further more, it's obtained that the number of the limit cycles of the perturbed system under quartic polynomial perturbations is no more than 7, and it is no more than 4 under cubic perturbations.