Abstract Our previous work (Xu et al. in Eur Phys J C 85(7):770, 2025) showed that the chaotic dynamics of charged test particles near a Kerr–MOG black hole depend on several parameters, such as the particle energy, angular momentum, magnetic field strength, black hole spin, and the MOG parameter. Yet, the influence of external electromagnetic fields on the long-term evolution of such particles cannot be neglected. To extend and refine this line of research, we generalize the electromagnetic four-potential in this work from the Wald form to a more general form. This extension is motivated by the fact that MOG theory generalizes the electromagnetic field solution – originally derived for vacuum, stationary, axisymmetric black holes in a uniform magnetic field – to non-vacuum backgrounds. Due to the resulting structural changes in the system’s Hamiltonian, we reconstruct a fourth-order explicit symplectic algorithm to ensure high accuracy and numerical stability in long-term integrations. Our results reveal that, compared to the Wald potential, the generalized potential substantially alters the chaotic dynamics by reducing the parameter threshold for the onset of chaos and expanding the chaotic domain in two-dimensional parameter space. These findings not only reaffirm the limitations of the Wald potential in non-vacuum environments but also support the generalized potential as a more suitable framework for studying chaotic phenomena in modified gravity.
Xu et al. (Sun,) studied this question.