Chaotic behavior of geodesics in Kerr-like spacetime

In recent years, experimental and theoretical research on the nature of supermassive bodies that inhabit central regions galaxies has grown significantly. this contribution, we study motion unitary mass test particles in a perturbed Kerr-like metric. The metric represents approximate exterior spacetime massive rotating body with quadrupole moment $q$. Through extensive geodesics, it is found chaotic behavior proliferates as $q$ parameter increases. Several structures are phase space, such Birkhoff chains islands hyperbolic points if $q\ensuremath{\ne}0$. Additionally, crossing null radial momentum axis (${p}_{r}=0$) studied by analyzing rotation number, several stability preserved deformation increases, while others destroyed. Furthermore, to be dictated stickiness, orbits remain attached stable ones for long time before they become divergent attracted event horizon.