Variational principle for the determination of unstable periodic orbits and instanton trajectories at saddle points

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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 the 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.