Chaotic dynamics of off-equatorial orbits around pseudo-Newtonian compact objects with dipolar halos

In this paper, we implement a generalised pseudo-Newtonian potential and prescribe a numerical fitting formalism to study the off-equatorial orbits inclined at a certain angle with the equatorial plane around both Schwarzschild and Kerr-like compact object primaries surrounded by a dipolar halo of matter. The chaotic dynamics of the orbits are detailed for both non-relativistic and special-relativistic test particles. The dependence of the degree of chaos on the rotation parameter $a$ and the inclination angle $i$ is established individually using widely used indicators, such as the Poincar\'e Map and the Lyapunov Characteristic Number. We find that although the chaoticity of the orbits has a positive correlation with $i$, the growth in the chaotic behaviour is not systematic. There exists a threshold value of the inclination angle $i_{\text{c}}$, after which the degree of chaos shows a sharp increase. On the other hand, the chaoticity of the inclined orbits anti-correlates with $a$ throughout its entire range. However, the negative correlation is systematic at lower values of the inclination angle. At higher values of $i$, the degree of chaos is maximum for the maximally counter-rotating compact objects, the Kerr parameter of which is below a threshold value $a_{\text{c}}$. Above this threshold value, the correlation becomes weak. Furthermore, we establish a qualitative correlation between the threshold values and the overall chaoticity. The studies performed with different orbital parameters and several initial conditions reveal the intricate nature of the system.