A Formal Series Approach to the Center Manifold Theorem

this paper considers near-equilibrium systems of ordinary differential equations with explicit separation of the slow and stable manifolds. formal b-series like those previously used to analyze highly oscillatory systems or to construct modified equations are employed here to construct expansions of the change of variables, the center invariant manifold and the reduced model. the new approach may be seen as a process of reduction to a normal form, with the surprising advantage, as compared to the standard view conveyed by the celebrated center manifold theorem, that it is possible to recover the complete solution at any time through an explicitly computable way.