In this work we consider the number of limit cycles that can bifurcate from
periodic annulus of the quintic Hamiltonian vector fields XH ¼ 2y @
@x 2xP0ðx2Þ @
@y
with P as an arbitrary polynomial of degree three under small perturbations of
the form "ð þ x2 þ x4Þy @
@y, where 05j"j  1 and ; ;  are real constants.
We show that the least upper bound for the number of limit cycles bifurcated
from the periodic annulus of XH surrounding the origin is two, when origin is
a global or local centre.