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

## TL;DR

This paper introduces a method using Lagrangian descriptors to directly construct unstable periodic orbits and instanton trajectories at saddle points in dynamical systems, simplifying their identification without extra constraints.

## Contribution

The paper presents a novel approach for directly constructing unstable periodic orbits and instanton trajectories at saddle points using Lagrangian descriptors, avoiding additional constraints.

## Key findings

- Successfully constructs periodic orbits at energies above the saddle point.
- Accurately determines instanton trajectories below the saddle point energy.
- Applicable to two-degree of freedom systems at rank-1 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.