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

The number of isolated zeros of the Melnikov function for a Hamiltonian system under polynomial perturbations is closed related to determinating the number of limit-cycles of a perturbated polynomial Hamiltonian system on the plane, so finding the upper bound of the number of isolated zeros of Melnikov function is an important problem in bifurcation theory of ordinary differential equation. It's studied that the number of isolated zeros of the Melnikov function for a one-parameter Cubic Hamiltonian system under polynomial perturbations in this paper. Consider the systemwhere b is a small parameter,∈is a perturbed parameter, 0<∈<<∈^?<1(h,b) and M₁⁺(h,b)(M₁^-(h,b)) were estimated by the expansion of Taylor about the small parameter b. When (?)M₁(h,b)|_b=0(?)(m is a non-negative integer), under the polynomial perturbation, the number of limit-cycles nearΓ_(h,b)={(x,y)|H(x,y)>0} is B(3,n)≤n+[(?)m]+1;when (?)M₁⁺(h,b)|_b=0(?)0 ((?)M₁^-M₁^-(h,b)|_b=0(?)0)(m is a non-negative integer), under the polynomial perturbation, the numberof limit-cycles nearΓ_(h,b)⁺={(x,y)|H(x,y)<0,x>0} (Γ_(h,b)^-={(x,y)|H(x,y)<0,x<0}) is B⁺(3,n)≤2n+7m+[(?)]+[(?)]+4(B^-(3,n)≤2n+7m+[(?)]+[(?)]+4). Further more, it's obtained that the number of the limit cycles of the perturbed system under cubic polynomial perturbations nearΓ_(h,b) is no more than 5, and nearΓ_(h,b)⁺ (Γ_(h,b)^-) is no more than 5 (5); the number of the limit cycles of the perturbed system under quartic polynomial perturbations nearΓ_(h,b) is no more than 11, and nearΓ_(h,b)⁺ (Γ_(h,b)^-) is no more than 9 (9).