Multiple timescales and the parametrisation method in geometric singular perturbation theory

The paper proves solvability and uniqueness of the infinitesimal conjugacy equations by projecting onto the tangent/fast splitting with an explicit projection P0 and yields closed formulas for ri and Yi. The candidate solution proves the same facts by constructing a smooth pointwise inverse of K = [Dφ0 N0], then applying the resulting left-inverses (A,B) to decouple the equations. Under the paper’s hypotheses (embedding, injective N0, invertible n0), both arguments are valid and equivalent. The model tacitly uses N0’s injectivity/transversality, which the paper states explicitly.