  In this paper we demonstrate the capability of the method of Lagrangian
descriptors to unveil the phase space structures that characterize transport in
high-dimensional symplectic maps. In order to illustrate its use, we apply it
to a four-dimensional symplectic map model that is used in chemistry to explore
the nonlinear dynamics of van der Waals complexes. The advantage of this
technique is that it allows us to easily and effectively extract the invariant
manifolds that determine the dynamics of the system under study by means of
examining the intersections of the underlying phase space structures with
low-dimensional slices. With this approach, one can perform a full
computational phase space tomography from which three-dimensional
representations of the higher-dimensional phase space can be systematically
reconstructed. This analysis may be of much help for the visualization and
understanding of the nonlinear dynamical mechanisms that take place in
high-dimensional systems. In this context, we demonstrate how this tool can be
used to detect whether the stable and unstable manifolds of the system
intersect forming turnstile lobes that enclose a certain phase space volume,
and the nature of their intersection.
