This is a joint work with Antonio Giorgilli and Ugo Locatelli.

In order to deal with lower dimensional tori, we first adapted our approach successfully used (in the past few years) to construct invariant maximal tori. In fact, we perform a sequence of normalization steps, which are based on the elimination of the perturbative terms through three canonical transformations similar to those introduced by Kolmogorov's in the classical KAM proof scheme. We think that this theoretical part of the work might extend also to the infinite-dimensional case, i.e. to some Hamiltonian PDE's.

The first application of our approach looks for an invariant elliptic torus not far from the real orbit of a planar planetary system including Sun, Jupiter, Saturn and Uranus (SJSU, for short). After some preliminary work (aiming to expand in canonical coordinates all the interaction terms between each pair
of planets appearing in the Hamiltonian of the SJSU system), we applied our constructive scheme. All these expansions are explicitly performed by algebraic manipulation on a computer. Finally, our semi-analytic procedure producing the motion on the elliptic torus is validated by using frequency analysis.