Abstract: We first examine the existence, uniqueness, regularity, twist and symplectic properties of compact invariant cylinders with boundary, located near simple or double resonances in perturbations of action-angle systems on the annulus $A^3$. We then prove they satisfy sufficient compatibility conditions on their dynamics and their homoclinic intersections, in order to prove the existence of drifting orbits along them, shadowing pseudo-orbits of inner-homoclinic polysystems. This provides us with a good control of the local behavior of the drifting orbits near essential hyperbolic 2-dimensional tori located inside the cylinders.