2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119530In this paper we prove that near the equilibirum position any periodic FPU chain with an odd number of particles admits a Birkhoff normal form up to order 4, and we obtain an explicit formula of the Hessian of its Hamiltonian at the fixed point.32 pages, corrected typosExactly Solvable and Integrable SystemsMathematical Physicstext