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

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