Librational KAM tori in the secular dynamics of the $\upsilon$ Andromedæ planetary system

Abstract

We study the planetary system of υ Andromedae, considering the three-body problem formed by the central star and the two largest planets, υ And c and υ And d. We adopt a secular, three-dimensional model and initial conditions within the range of the observed values. The numerical integrations highlight that the system is orbiting around a one-dimensional elliptic torus (i.e., a periodic orbit that is linearly stable). This invariant object is used as a seed for an algorithm based on a sequence of canonical transformations. The algorithm determines the normal form related to a KAM torus, whose shape is in excellent agreement with the orbits of the secular model. We rigorously prove that the algorithm constructing the final KAM invariant torus is convergent, by adopting a suitable technique based on a computer-assisted proof.