Numerical integration of variational equations

Ch. Skokos and E. Gerlach
Phys. Rev. E 82, 036704 – Published 30 September 2010

Abstract

We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom and investigate their efficiency in accurately reproducing well-known properties of chaos indicators such as the Lyapunov characteristic exponents and the generalized alignment indices. We find that the best numerical performance is exhibited by the “tangent map method,” a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton equations of motion by the repeated action of a symplectic map S, while the corresponding tangent map TS is used for the integration of the variational equations. A simple and systematic technique to construct TS is also presented.

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  • Received 1 June 2010

DOI:https://doi.org/10.1103/PhysRevE.82.036704

©2010 American Physical Society

Authors & Affiliations

Ch. Skokos1 and E. Gerlach2

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

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Issue

Vol. 82, Iss. 3 — September 2010

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