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

Abstract

The complexity of arbitray 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 scalarvalued 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.

Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon-Heiles-type potential

“…where p ∈ (0, 1] and τ ∈ R + are freely chosen parameters, and the overdot symbol represents the derivative with respect to time. It is to be noted here that there are three formulations of the function M p in the literature: the arc length of a trajectory in phase space [36], the arc length of a trajectory projected on the configuration space [47,[58][59][60], and the sum of the p-norm of the vector field components [39,61]. Although the latter formulation of the Lagrangian descriptor (10) developed in Ref.…”

Finding NHIM: Identifying high dimensional phase space structures in reaction dynamics using Lagrangian descriptors

“…where p ∈ (0, 1] and τ ∈ R + are freely chosen parameters, and the overdot symbol represents the derivative with respect to time. It is to be noted here that there are three formulations of the function M p in the literature: the arc length of a trajectory in phase space [33], the arc length of a trajectory projected on the configuration space [26,27,28,29], and the sum of the p-norm of the vector field components [31,32]. Although the latter formulation of the Lagrangian descriptor (2) developed in Ref.…”

“…Recently the applicability of the Lagrangian descriptor based approach to time dependent problems with random and dissipative forcing in chemical reaction dynamics has been shown whereby the transition state trajectory is calculated using the extremal values in Lagrangian descriptor values; see Refs. [4,5,6,15,26,27,28,29,39,41]. The initial conditions for the transition state trajectory is identified by computing the extrema of the Lagrangian descriptor on a two dimensional domain.…”