Sampling Phase Space Dividing Surfaces Constructed from Normally Hyperbolic Invariant Manifolds (NHIMs)

In this paper, we further investigate the construction of a phase space dividing surface (DS) from a normally hyperbolic invariant manifold (NHIM) and the sampling procedure for the resulting dividing surface described in earlier work ( Wiggins , S. ; J. Chem. Phys. 2016 , 144 , 054107 ). Our discussion centers on the relationship between geometrical structures and dynamics for 2 and 3 degree of freedom (DoF) systems, specifically, the construction of a DS from a NHIM. We show that if the equation for the NHIM and associated DS is known (e.g., as obtained from Poincaré-Birkhoff normal form theory), then the numerical procedure described in Wiggins et al. ( J. Chem. Phys. 2016 , 144 , 054107 ) gives the same result as a sampling method based upon the explicit form of the NHIM. After describing the sampling procedure in a general context, it is applied to a quadratic Hamiltonian normal form near an index-one saddle where explicit formulas exist for both the NHIM and the DS. It is shown for both 2 and 3 DoF systems that a version of the general sampling procedure provides points on the analytically defined DS with the correct microcanonical density on the constant-energy DS. Excellent agreement is obtained between analytical and numerical averages of quadratic energy terms over the DS for a range of energies.