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

Chaotic von Zeipel-Lidov-Kozai Oscillations of Binary System around Rotating Supermassive Black Hole