Comparing the efficiency of numerical techniques for the integration of variational equations

TL;DR

This paper compares various numerical techniques for integrating variational equations in Hamiltonian systems, assessing their accuracy and efficiency using the Hénon-Heiles system and chaos detection methods.

Contribution

It provides a systematic comparison of numerical methods for variational equations in Hamiltonian systems, highlighting their relative efficiency and accuracy.

Findings

Certain methods outperform others in speed and accuracy.

The choice of technique affects chaos detection results.

Efficiency varies depending on the specific numerical scheme used.

Abstract

We present a comparison of different numerical techniques for the integration of variational equations. The methods presented can be applied to any autonomous Hamiltonian system whose kinetic energy is quadratic in the generalized momenta, and whose potential is a function of the generalized positions. We apply the various techniques to the well-known H\'enon-Heiles system, and use the Smaller Alignment Index (SALI) method of chaos detection to evaluate the percentage of its chaotic orbits. The accuracy and the speed of the integration schemes in evaluating this percentage are used to investigate the numerical efficiency of the various techniques.