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The very long-term evolution of the hierarchical restricted three-bodyproblem with a slightly aligned precessing quadrupole potential is studiedanalytically. This problem describes the evolution of a star and a planet whichare perturbed either by a (circular and not too inclined) binary star system orby one other star and a second more distant star, as well as a perturbation byone distant star and the host galaxy or a compact-object binary system orbitinga massive black hole in non-spherical nuclear star clusters\citep{hamers2017,petrovich2017}. Previous numerical experiments have shownthat when the precession frequency is comparable to the Kozai-Lidov time scale,long term evolution emerges that involves extremely high eccentricities withpotential applications for a broad scope of astrophysical phenomena includingsystems with merging black holes, neutron stars or white dwarfs. By averagingthe secular equations of motion over the Kozai-Lidov Cycles (KLCs) we solve theproblem analytically in the neighborhood of the KLC fixed point where theeccentricity vector is close to unity and aligned with the quadrupole axis andfor a precession rate similar to the Kozai Lidov time scale. In this regime thedynamics is dominated by a resonance between the perturbation frequency and theprecession frequency of the eccentricity vector. While the quantitativeevolution of the system is not reproduced by the solution far away from thisfixed point, it sheds light on the qualitative behaviour.

Analytic understanding of the resonant nature of Kozai Lidov Cycles with a precessing quadrupole potential