The Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems with Three or More Degrees of Freedom - II

We develop a method for the construction of a dividing surface using periodic orbits in Hamiltonian systems with three or more degrees-of-freedom that is an alternative to the method presented in [Katsanikas & Wiggins, 2021]. Similar to that method, for an n degrees-of-freedom Hamiltonian system, we extend a one-dimensional object (the periodic orbit) to a 2n- 2 dimensional geometrical object in the energy surface of a 2n- 1 dimensional space that has the desired properties for a dividing surface. The advantage of this new method is that it avoids the computation of the normally hyperbolic invariant manifold (NHIM) (as the first method did) and it is easier to numerically implement than the first method of constructing periodic orbit dividing surfaces. Moreover, this method has less strict required conditions than the first method for constructing periodic orbit dividing surfaces. We apply the new method to a benchmark example of a Hamiltonian system with three degrees-of-freedom for which we are able to investigate the structure of the dividing surface in detail. We also compare the periodic orbit dividing surfaces constructed in this way with the dividing surfaces that are constructed starting with a NHIM. We show that these periodic orbit dividing surfaces are subsets of the dividing surfaces that are constructed from the NHIM.