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We present a method that generalizes the periodic orbit dividing surfaceconstruction for Hamiltonian systems with three or more degrees of freedom. Weconstruct 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 presentour methods using benchmark examples for two and three degree of freedomHamiltonian systems to illustrate the corresponding algorithm for thisconstruction. Towards this end we use the normal form quadratic Hamiltoniansystem with two and three degrees of freedom. We found that the periodic orbitdividing surface can provide us the same dynamical information as the dividingsurface constructed using normally hyperbolic invariant manifolds. This issignificant because, in general, computations of normally hyperbolic invariantmanifolds are very difficult in Hamiltonian systems with three or more degreesof freedom. However, our method avoids this computation and the onlyinformation that we need is the location of one periodic orbit.

The Generalization of the Periodic Orbit Dividing Surface in Hamiltonian Systems with three or more degrees of freedom -- I