A Rigidity Property of Perturbations of n Identical Harmonic Oscillators

Let Xε: S2n-1→ TS2n-1 be a smooth perturbation of X, the vector field associated to the dynamical system defined by n identical uncoupled harmonic oscillators constrained to their 1-energy level. We are dealing with the case when any orbit of every Xε is closed: while in general is false that the vector fields of the perturbation are orbitally equivalent to the unperturbed X (Villarini in Ergod Theory Dyn Syst 39:1–32, 2019), we prove that this rigidity behaviour is indeed true if each Xε restricted to a codimension 2 sphere in S2n-1 is orbitally conjugated to a subsystem of X made by n- 1 harmonic oscillators. In other words: to have a non-rigid, or truly non-linear, perturbation of X at least two harmonic oscillators must be destroyed by the perturbation. We use this rigidity result to prove a linearization theorem for real analytic multicentres. Finally we give an example of a real analytic perturbation of X showing discontinuous changing of integer invariants of the vector fields of the perturbation.