  We present a method that generalizes the periodic orbit dividing surface
construction for Hamiltonian systems with three or more degrees of freedom. We
construct a torus using as a basis a periodic orbit and we extend this to a
$2n-2$ dimensional object in the $2n-1$ dimensional energy surface. We present
our methods using benchmark examples for two and three degree of freedom
Hamiltonian systems to illustrate the corresponding algorithm for this
construction. Towards this end we use the normal form quadratic Hamiltonian
system with two and three degrees of freedom. We found that the periodic orbit
dividing surface can provide us the same dynamical information as the dividing
surface constructed using normally hyperbolic invariant manifolds. This is
significant because, in general, computations of normally hyperbolic invariant
manifolds are very difficult in Hamiltonian systems with three or more degrees
of freedom. However, our method avoids this computation and the only
information that we need is the location of one periodic orbit.
