The Number Of Limit-cycles Of A Class Of Hamiltonian System Under Polynomial Perturbation

Finding the upper bound for the number of isolated zeros of Mel'nikov function is an important problem of bifurcation theory of ordinary differential equation. It is closed related to determinating the number of limit-cycles of a perturbated polynomial Hamiltonian system on the plane. It has been studied in this paper that the number of isolated zeros of Mel'nikov function for a class of the Hamiltonian system with double-homoclinic orbit, under polynomial perturbation. Consider the system:where 0≤ε(?)1 is a small parameter, P_n(x, y) and Q_n(x, y)are polynomials of degrees not greater than n. Whenε=0, the system is a Hamiltonian system which exists a double-homoclinic orbit,Γ₀ = {(x,y)|H(x,y) =1/2y²-1/2x² + 1/4x⁴ = 0}. It has been discussed in this paper that the bifurcation phenomenon of the family of the orbits outsideΓ₀,Γ_h= {(x, y)|H(x, y) > 0}, under polynomial perturbation. When the Petrov's method was used to estimate the number of isolate zeros of Mel'nikov function, there was a singular point on the branch cut. The branch cut has been improved in the neighborhood of the critical point. The number of limit-cycles of the perturbed system is no more than n. Then, it has been considered in this paper that the bifurcation phenomenon of two symmetry familys of orbits insideΓ₀,Γ_h⁺= {(x,y)|H(x,y) <0,x >},Γ_h^- = {(x,y)|H{x,y) <0,x < 0}, under polynomial perturbation. The algebra structure of M₁(h) iswhere degg₀^*(h)≤[(n-1)/2], degg₂^*(h)≤[(n-3)/2], degg(h)≤[n/2].When the M₁(h) was derivated [n/2] + 1 times, g(h) was deleted.degG_0,[n/2]+1≤n, degG(2,[n/2]+1)≤n-1. The number of limit-cycles of the perturbedsystem are no more than 2n+[n/2]+2 in the neighborhood ofΓ_h⁺ andΓ_h^-, applyingPerov's method.