  The complexity of arbitrary dynamical systems and chemical reactions, in
particular, can often be resolved if only the appropriate periodic orbit - in
the form of a limit cycle, dividing surface, instanton trajectories or some
other related structure - can be uncovered. Determining such a periodic orbit,
no matter how beguilingly simple it appears, is often very challenging. We
present a method for the direct construction of unstable periodic orbits and
instanton trajectories at saddle points by means of Lagrangian descriptors.
Such structures result from the minimization of a scalar-valued phase space
function without need for any additional constraints or knowledge. We
illustrate the approach for two-degree of freedom systems at a rank-1 saddle
point of the underlying potential energy surface by constructing both periodic
orbits at energies above the saddle point as well as instanton trajectories
below the saddle point energy.
