Invariant manifolds near a minimizer

A classical result, studied, among others, by Carathéodory [C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, Chelsea, New York, 1989], states that, for second-order, scalar equations, nondegenerate periodic minimizers are hyperbolic. Consequently, the Stable/Unstable Manifold Theorem applies, and implies that, at least locally, the stable and unstable sets are regular curves intersecting transversally at the nondegenerate minimizer. For analytic equations, there is a version of this fact which holds for isolated, but possibly degenerate, minimizers.