  It this paper we study a class of perturbed Hamiltonian systems under
perturbations of thirteen order in order to detect the number of limit cycles
which bifurcate from some periodic orbits of the unperturbed Hamiltonian
system. The system has been previously studied in \cite{tig2}, \cite{tig3},
\cite{tig4}. We observe in the present work that the system under perturbations
of thirteen order can have more limit cycles than under perturbations of five
\cite{tig3}, respectively nine \cite{tig4} order but we have not identified
more limit cycles than under perturbations of seven \cite{tig2} order.
