Abstract

In this work, we consider a generic 2-DOF autonomous Hamiltonian system, which linear part has all eigenvalues equal to zero. For these type of Hamiltonian, we provide, in an explicit and constructive way, the normal form up to the term of the fourth order. Additionally, we analyze the stability of the normalized Hamiltonian in several cases. Precisely, we obtain instability of the truncated Hamiltonian by explicitly building an invariant ray solution.