On rigorous modified equations for discretizations of ODEs

We construct in a novel way the modified vector field used in backward error analysis of discretizations of ODEs. In addition to giving improvements of known results, the new result includes a small, analytic time-dependent perturbation so that the the flow of the modified vector field exactly reproduces the numerical approximations produced by one-step methods. This simplifies analysis of resonance effects prevalent in conservative discretizations, and also paves the way to easy proofs of stability results such as the KAM theorem, which previously was proved by ad-hoc methods for numerical methods.