A family of symplectic integrators adapted for the integration of perturbed
Hamiltonian systems of the form H = A + epsilonB was given in (McLachlan,
1995). We give here a constructive proof that for all integer p, such integ
rator exists, with only positive steps, and with a remainder of order O(tau
(p)epsilon + tau (2)epsilon (2) ), where tau is the stepsize of the integr
ator. Moreover, we compute the analytical expressions of the leading terms
of the remainders at all orders. We show also that for a large class of sys
tems, a corrector step can be performed such that the remainder becomes O(t
au (p)epsilon + tau (4)epsilon (2)). The performances of these integrators
are compared for the simple pendulum and the planetary three-body problem o
f Sun-Jupiter-Saturn.