Secular dynamics of the three body problem: application to the upsilon Andromedae planetary system

(abbreviated) We use a semi-numerical approach to study the secular behavior of a system composed of a central star and two massive planets in eccentric co-planar orbits. We show that the secular dynamics of this system can be described using only two parameters, the ratios of the semimajor axes and the planetary masses. The main dynamical features of the system are presented in geometrical pictures that allows us to investigate a large domain of the secular phase space of this three-body problem without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. We show that three different regimes of secular motion with respect to the secular angle $\Delta\varpi$ are possible: they are circulation, oscillation (around 0 and $180^\circ$), and high eccentricity libration in a nonlinear secular resonance. The first two regimes are a continuous extension of the classical linear secular perturbation theory; the last is a new feature, hitherto unknown, in the secular dynamics of the three-body problem.