THE NUMBER OF LIMIT CYCLES FROM ELLIPTIC HAMILTONIAN VECTOR FIELDS BY HIGHER ORDER MELNIKOV FUNCTIONS

Abstract

In this paper, the perturbed Hamiltonian system $dH=\epsilon F_4+\epsilon^2F_3+\epsilon^3F_2+\epsilon^4F_1$, with $F_i$ the vector valued homogeneous polynomials of degree $i$. The Hamiltonian function is $H=y^2/2+U(X),$ where $U$ is a univariate polynomial of degree four without symmetry. By computing higher order Melnikov functions, the upper bounds for the number of limit cycles that bifurcate from $dH=0$ are deserved.