Long-Time Accuracy for Approximate Slow Manifolds in a Finite-Dimensional Model of Balance

We study the slow singular limit for planar anharmonic oscillatory motion of a charged particle under the influence of a perpendicular magnetic field when the mass of the particle goes to zero. This model has been used by the authors as a toy model for exploring variational high-order approximations to the slow dynamics in rotating fluids. In this paper, we address the long time validity of the slow limit equations in the simplest nontrivial case. We show that the first-order reduced model remains O(ε) accurate over a long 1/ε timescale. The proof is elementary, but involves subtle estimates on the nonautonomous linearized dynamics.