Testing Particle Dynamics, Epicyclic Oscillations around Rotating Bumblebee Kerr-Newman-AdS Black Hole

In this study, we investigate the dynamics of test particles in the spacetime of rotating Bumblebee Kerr-Newman-Anti-de Sitter (AdS) black hole (BH) solutions, in which Lorentz symmetry violation is incorporated through the bumblebee gravity framework. We first derive the geodesic equations that govern particle trajectories and determine the conditions for the existence of stable circular orbits in this modified rotating geometry. In addition, we obtain the exact form of the effective potential, which facilitates a quantitative analysis of orbital stability and elucidates the influence of the bumblebee parameter on particle motion in the vicinity of the BH. Subsequently, we analyze small perturbations about circular orbits to study linearized oscillations in both the radial and vertical directions. Within this perturbative scheme, we compute the corresponding epicyclic frequencies and express them in a form suitable for interpretation by a distant observer. In this context, our results provide a systematic characterization of oscillatory motion in the rotating Bumblebee Kerr-Newman-AdS spacetime and demonstrate how Lorentz-violating contributions modify both the orbital dynamics and the associated frequency spectrum. These findings are pertinent to the theoretical modeling of high-energy astrophysical phenomena, particularly quasi-periodic oscillations (QPOs) observed in accreting BH systems.