Quantitative estimates on the normal form around a non semi-simple 1:-1 resonant periodic orbit

Abstract

The purpose of thiswork is to give precise estimates for the size of the remainder of the normalized Hamiltonian around a non-semi-simple 1 : −1 resonant periodic orbit, as a function of the distance to the orbit. We consider a periodic orbit of a real analytic three-degrees of freedom Hamiltonian system having a pairwise collision of its non-trivial characteristic multipliers on the unit circle. Under generic hypotheses of non-resonance and non-degeneracy of the collision, we present a constructive methodology to reduce the Hamiltonian around the orbit to its (integrable) normal form, up to any given order. This constructive process allows to obtain quantitative estimates for the size of the remainder of the normal form, as a function of the normalizing order. By selecting appropriately this order in terms of the distance R to the resonant orbit (measured using suitable coordinates), r = ropt(R) := 2 + ?exp(W(log(1/R1/(τ+1+ε))))?, we have proved that the size of the remainder can be bounded (for small R) by Rropt(R)/2. Here, W(·) stands for Lambert’s W function and verifies that W(z) exp(W(z)) = z, τ ? 1 is the exponent of the required Diophantine condition and ε > 0 is any small quantity. The reasons leading to this bound instead of classical exponentially small estimates are also discussed.