Examining the classification of orbits within the context of the axisymmetric restricted five-body problem

This study aims to provide a thorough numerical investigation of the orbital dynamics involved in the axisymmetric restricted problem of five bodies. We obtain the basins of escape by numerically integrating various large sets of initial conditions of orbits. Further, the influence of various perturbation parameters, mainly the angle parameters, on the orbital structure of the system in three different cases of the axisymmetric restricted five-body problem, i.e., (i) the convex configuration, (ii) the first concave configuration, and (iii) the second concave case. In all these cases, we have classified the nature of the motion of the test particle into mainly three brackets: (i) the collisional orbit between the orbiting test particle and one of the primaries, (ii) the bounded regular orbits, and (iii) the escaping orbits. We also determined the collisional and escape basins and evaluated the associated collision and escape times. The collisional orbits, measured on the color-coded diagrams, clearly exhibit a dependence on the angle parameters. The highly fractal basin boundaries have a sensitive relation with the initial conditions, which implies an uncertainty among escape solutions that unfold to different regions of phase space. Moreover, the characteristics of escape basins are strongly related to the orbital energy, with an astonishing presence of fractal basin boundaries along all the escape regimes.