  In Hamiltonian systems subjected to periodic perturbations the stable and
unstable manifolds of the unstable periodic orbits provide the dynamical
"skeleton" that drives the mixing process and bounds the chaotic regions of the
phase space. Determining the behavior of these objects is valuable in physical
applications involving asymmetric solenoidal fields or time-dependent
Hamiltonian systems. Here we introduce a simple method to calculate an unstable
periodic orbit given an initial guess on its position. Then we present an
efficient adaptive method to build its high-resolution invariant manifolds to
arbitrary length and compare it to a random sampling method with the same
computational cost. The adaptive method gives a high-quality representation of
the manifolds and reveals fine details that become lost in the random sampling
method. Finally, we introduce an approximation to the adaptive method to build
the manifolds avoiding redundant calculations and reducing logarithmically the
number of computations needed to represent these surfaces.
