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

Abstract

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.

5 pages, 3 figures, accepted for publication in Phys. Rev. A

References in corpus (10)

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Cited by in corpus (2)

• Finding NHIM: Identifying High Dimensional Phase Space Structures in Reaction Dynamics using Lagrangian Descriptors
• Finding NHIM in 2 and 3 degrees-of-freedom with Hénon-Heiles type potential