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In this paper we compare the method of Lagrangian descriptors with theclassical method of Poincare maps for revealing the phase space structure oftwo degree-of-freedom Hamiltonian systems. The comparison is carried out byconsidering the dynamics of a two degree-of-freedom system having a valleyridge inflection point (VRI) potential energy surface. VRI potential energysurfaces have four critical points: a high energy saddle and a lower energysaddle separating two wells. In between the two saddle points is a valley ridgeinflection point that is the point where the potential energy surface geometrychanges from a valley to a ridge. The region between the two saddles forms areaction channel and the dynamical issue of interest is how trajectories crossthe high energy saddle, evolve towards the lower energy saddle, and select aparticular well to enter. Lagrangian descriptors and Poincare maps are comparedfor their ability to determine the phase space structures that govern thisdynamical process.

From Poincare Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential