Alessandra Celletti

The Galilean satellites of Jupiter are called Io, Europa, Ganymede and Callisto. The first three moons are found in the so-called Laplace resonance, which means that their orbits are locked in a 2 : 1 resonant chain. Dissipative tidal effects play a fundamental role, especially when considered on long timescales. The main objective of this work is the study of the persistence of the resonance along the evolution of the system when considering the tidal interaction between Jupiter and Io. To constrain the computational cost of the task, we enhance this dissipative effect by means of a multiplying factor. We develop a simplified model to study the propagation of the tidal effects from Io to the other moons, resulting in the outward migration of the satellites. We provide an analytical description of the phenomenon, as well as the behaviour of the semi-major axis of Io as a function of the figure of merit. We also consider the interaction of the inner trio with Callisto, using a more elaborated Hamiltonian model allowing us to study the long-term evolution of the system along few gigayears. We conclude by studying the possibility of the trapping into resonance of Callisto depending on its initial conditions.

We consider a dissipative vector field which is represented by a nearly-integrable Hamiltonian flow to which a dissipative contribution is added. The vector field depends upon two parameters, namely the perturbing and dissipative parameters, and by a drift term. We study an $\mathcal{l}$-dimensional, time-dependent vector field, which is motivated by mathematical models in Celestial Mechanics. Assuming to start with non-resonant initial conditions, we provide the construction of the normal form up to an arbitrary order. To construct the normal form, a suitable choice of the drift parameter must be performed. The normal form allows also to provide an explicit expression of the frequency associated to the normalized coordinates. We also give an example in which we construct explicitly the normal form, we make a comparison with a numerical integration, and we determine the parameter values and the time interval of validity of the normal form.

We consider nearly-integrable systems under a relatively small dissipation. In particular we investigate two specific models: the discrete dissipative standard map and the continuous dissipative spin-orbit model. With reference to such samples, we review some analytical and numerical results about the persistence of invariant attractors and of periodic attractors.

We investigate the break-down of invariant tori in a four dimensional standard mapping for different values of the coupling parameter. We select various two-dimensional 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 break-down threshold of the invariant tori. In particular, the analyticity domain is recovered by means of a suitable implementation of Padé approximants. The break-down 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.

Birkhoff periodic orbits associated to spin-orbit resonances in Celestial Mechanics and in particular to the Moon–Earth and Mercury–Sun systems are considered. A general method (based on a quantitative version of the Implicit Function Theorem) for the construction of such orbits with particular attention to "effective estimates" on the size of the perturbative parameters is presented and tested on the above mentioned systems. Lyapunov stability of the periodic orbits (for small values of the perturbative parameters) is proved by constructing KAM librational invariant surfaces trapping the periodic orbits.