  In chemical or physical reaction dynamics, it is essential to distinguish
precisely between reactants and products for all time. This task is especially
demanding in time-dependent or driven systems because therein the dividing
surface (DS) between these states often exhibits a nontrivial time-dependence.
The so-called transition state (TS) trajectory has been seen to define a DS
which is free of recrossings in a large number of one-dimensional reactions
across time-dependent barriers, and, thus, allows one to determine exact
reaction rates. A fundamental challenge to applying this method is the
construction of the TS trajectory itself. The minimization of Lagrangian
descriptors (LDs) provides a general and powerful scheme to obtain that
trajectory even when perturbation theory fails. Both approaches encounter
possible breakdowns when the overall potential is bounded, admitting the
possibility of returns to the barrier long after trajectories have reached the
product or reactant wells. Such global dynamics cannot be captured by
perturbation theory. Meanwhile, in the LD-DS approach, it leads to the
emergence of additional local minima which make it difficult to extract the
optimal branch associated with the desired TS trajectory. In this work, we
illustrate this behavior for a time-dependent double-well potential revealing a
self-similar structure of the LD, and we demonstrate how the reflections and
side-minima can be addressed by an appropriate modification of the LD
associated with the direct rate across the barrier.
