High order symplectic integrators for perturbed Hamiltonian systems

We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form $H=A+ϵB$. We give a constructive proof that for all integer $p$, there exists an integrator with positive steps with a remainder of order $O({\tau }^pϵ+{\tau }^2{ϵ}^2)$, where $\tau$ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given at all orders. In many cases, a corrector step can be performed such that the remainder becomes $O({\tau }^pϵ+{\tau }^4{ϵ}^2)$. The performances of these integrators are compared for the simple pendulum and the planetary 3-Body problem of Sun-Jupiter-Saturn.