Abstract:

Given a polynomial $F=p circ H$, where $p$ is a polynomial in one variable of degree at least 2, and $H$ is a polynomial in two variables, we consider small polynomial perturbations of the corresponding degenerated Hamiltonian system. We will study the structure of the Poincaré-Pontryagin-Melnikov functions associated to the perturbed degenerated Hamiltonian system. We apply the result to the case $H=(x^2+y^2)/2$ in order to study the maximum number of zeros of the first Poincaré-Pontryagin-Melnikov function that does not vanish identically.