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

Enrico Gerlach; Charlampos Skokos

doi:10.3934/proc.2011.2011.475

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2011, Volume 2011: 475-484. Doi: 10.3934/proc.2011.2011.475 open access

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

Enrico Gerlach^1, and
Charlampos Skokos^2,

1.
Lohrmann Observatory, Technical University Dresden, D-01062 Dredsen
2.
Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, D-01187 Dresden

Received: July 2010

Revised: April 2011

Published: August 2017

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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.

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Mathematics Subject Classification: Primary: 37M25, 65P10; Secondary: 70H07, 37J99, 65P20.

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