Finding NHIM: Identifying high dimensional phase space structures in reaction dynamics using Lagrangian descriptors

Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, and their stable and unstable manifolds have been an integral part of computing quantitative results such as the transition fraction, rate of stability basin erosion in multi-stable mechanical systems, and rate constants in chemical reactions. Thus, methods that can reveal the geometry of these invariant manifolds in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with an index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor contour maps and the phase space structures’ intersection with an isoenergetic two-dimensional surface, and hence provides a verification of the method.