  In this paper we compare the method of Lagrangian descriptors with the
classical method of Poincare maps for revealing the phase space structure of
two degree-of-freedom Hamiltonian systems. The comparison is carried out by
considering the dynamics of a two degree-of-freedom system having a valley
ridge inflection point (VRI) potential energy surface. VRI potential energy
surfaces have four critical points: a high energy saddle and a lower energy
saddle separating two wells. In between the two saddle points is a valley ridge
inflection point that is the point where the potential energy surface geometry
changes from a valley to a ridge. The region between the two saddles forms a
reaction channel and the dynamical issue of interest is how trajectories cross
the high energy saddle, evolve towards the lower energy saddle, and select a
particular well to enter. Lagrangian descriptors and Poincare maps are compared
for their ability to determine the phase space structures that govern this
dynamical process.
