  Consider the spatial three-body problem, in the regime where one body
revolves far away around the other two, in space, the masses of the bodies
being arbitrary but fixed; in this regime, there are no resonances in mean
motions. The so-called secular dynamics governs the slow evolution of the
Keplerian ellipses. We show that it contains a horseshoe and all the chaotic
dynamics which goes along with it, corresponding to motions along which the
eccentricity of the inner ellipse undergoes large, random excursions. The proof
goes through the surprisingly explicit computation of the homoclinic solution
of the first order secular system, its complex singularities and the Melnikov
potential.
