Ugo Locatelli
via della Ricerca Scientifica 1, 00133, Roma, Italy
Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”
Publications:
Giorgilli A., Locatelli U., Sansottera M.
Secular Dynamics of a Planar Model of the SunJupiterSaturnUranus System; Effective Stability in the Light of Kolmogorov and Nekhoroshev Theories
2017, vol. 22, no. 1, pp. 5477
Abstract
We investigate the longtime stability of the SunJupiterSaturnUranus system by considering a planar secular model, which can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that underlies the analytic part of Nekhoroshev’s theorem to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller.

Locatelli U., Giorgilli A.
Construction of Kolmogorov's normal form for a planetary system
2005, vol. 10, no. 2, pp. 153171
Abstract
We describe an algorithm constructing an invariant KAM torus for a class of planetary systems, such that the mutual attractions, the eccentricities and the inclinations of the planets are small enough. By using computer algebra, we explicitly implement this algorithm for approximating a KAM torus for the problem of three bodies in a case similar to the Sun–Jupiter–Saturn system. We show that, by reducing the masses of the planets by a factor 10 and with a small displacement of the orbits, our semianalytical construction of the torus turns out to be successful.

Celletti A., Falcolini C., Locatelli U.
On the breakdown threshold of invariant tori in four dimensional maps
2004, vol. 9, no. 3, pp. 227253
Abstract
We investigate the breakdown of invariant tori in a four dimensional standard mapping for different values of the coupling parameter. We select various twodimensional frequency vectors, having eventually one or both components close to a rational value. The dynamics of this model is very reach and depends on two parameters, the perturbing and coupling parameters. Several techniques are introduced to determine the analyticity domain (in the complex perturbing parameter plane) and to compute the breakdown threshold of the invariant tori. In particular, the analyticity domain is recovered by means of a suitable implementation of Padé approximants. The breakdown threshold is computed through a suitable extension of Greene's method to four dimensional systems. Frequency analysis is implemented and compared with the previous techniques.
