  In this paper, we investigate the dynamics of a binary system that orbits a
rotating supermassive black hole. Our approach employs Fermi-Walker transport
to construct a local inertial reference frame, and to set up a Newtonian binary
system. We consider a scenario in which a circular geodesic observer is
positioned around a Kerr black hole, and thereby derive the equations of motion
governing the binary system. To eliminate the interaction terms between the
center of mass (CM) of the binary and its relative coordinates, we introduce a
small acceleration for the observer. This adjustment leads to the CM closely
following the observer's orbit, deviating from a circular geodesic. Here, we
first focus on elucidating the stability conditions in a hierarchical triple
system. Subsequently, we discuss the phenomenon of von Zeipel-Lidov-Kozai
(vZLK) oscillations, which manifest when the binary system is compact and the
initial inclination exceeds a critical angle. In hard binary systems, these
oscillations exhibit regular behavior, while in soft binary systems, they
exhibit a chaotic character, characterized by irregular periods and amplitudes,
albeit remaining stable. Additionally, we observe an orbital flip under
circumstances of large initial inclination. As for the motion of the CM, we
observe deviations from a purely circular orbit that transform into stable yet
chaotic oscillations characterized by minute amplitude variations.
