Abstract
We investigate the basins of convergence associated with the equilibrium points in the Sitnikov five-body problem with oblate primaries using Newton’s iteration method. Initially, we analyze the equilibrium positions as a function of the oblateness factor \(\sigma \) , finding that the nature of these equilibria changes with varying \(\sigma \) . Various cases are considered to study the behavior of these equilibrium positions. Subsequently, we graphically illustrate the effect of the oblateness factor \(\sigma \) on the basins of convergence related to the equilibrium positions in the complex plane. Specifically, for a given oblateness factor \(\sigma \) , the convergence region around the equilibrium points \({{E}_{i}}\) \((i = 0,1,2,3,4)\) is finite. When \(\sigma < 0\) , the convergence region around these points decreases as \(\sigma \) increases. Conversely, when \(\sigma > 0\) , the convergence region increases with \(\sigma \) . We develop a series solution to the problem using the Green’s function approach. For \({{\sigma }_{1}} < \sigma < \sigma _{2}^{*}\) , \({{\sigma }_{2}} < \sigma < {{\sigma }_{3}}\) and \(\sigma > {{\sigma }_{3}}\) , the infinitesimal mass exhibits perpetual periodicity with consistent cyclic behavior over time. In contrast, for \(\sigma < {{\sigma }_{1}}\) and \(\sigma _{2}^{*} < \sigma < {{\sigma }_{2}}\) , the motion of the infinitesimal mass grows exponentially. Our comprehensive examination aims to provide valuable insights to enhance scientific understanding in these areas. Ultimately, this study may support future research on convergent systems influenced by factors such as oblateness.
