Secular instability in the spatial three-body problem
Résumé
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 arbi-
trary 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, corre-
sponding 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.