Study of the sequences of bifurcations in the spatial octupolar Hamiltonian model of the planetary three body problem

The thesis is devoted to investigate the differences between two distinct approximations of the hamiltonian modelling a three- dimensional planetary system. In particular when dealing with secular models of the three- body problem a so- called quadrupolar approximation of such a dynamical system yealds an integrable hamiltonian, whose dynamic produces a single periodic orbit. On the other hand, an higher order approximation, we will present in wich sense during the thesis, called octupolar approximation, represent no more an integrable system and its dynamics produces two distinct periodic orbits. The aim of the thesis is to investigate how this bifurcations take places, refferring, as numerical model, to a specific exoplanetary system, or better to a toy model of it, called nu Andromedae. In particular we will try to apply tecniques that we use in the integrable case to an approximation of the non integrable system obtained through normal forms.