Numerical integration of high-order variational equations of ODEs

This paper discusses the numerical integration of high-order variational equations of ODEs. It is proved that, given a numerical method (say, any Runge–Kutta or Taylor method), to use automatic differentiation on this method (that is, using jet transport up to order p with a time step h for the numerical integration) produces exactly the same results as integrating the variational equations up to of order p with the same method and time step h as before. This allows to design step-size control strategies based on error estimates of the orbit and of the jets. Finally, the paper discusses how to use jet transport to obtain power expansions of Poincaré maps (either with spatial or temporal Poincaré sections) and invariant manifolds. Some examples are provided.