Numerical integration of variational equations
  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 like the Lyapunov
Characteristic Exponents (LCEs) and the Generalized Alignment Indices (GALIs).
We find that the best numerical performance is exhibited by the
\textit{`tangent map (TM) 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's 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.
