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The circular Sitnikov problem, where the two primary bodies are prolate oroblate spheroids, is numerically investigated. In particular, the basins ofconvergence on the complex plane are revealed by using a large collection ofnumerical methods of several order. We consider four cases, regarding the valueof the oblateness coefficient which determines the nature of the roots(attractors) of the system. For all cases we use the iterative schemes forperforming a thorough and systematic classification of the nodes on the complexplane. The distribution of the iterations as well as the probability and theircorrelations with the corresponding basins of convergence are also discussed.Our numerical computations indicate that most of the iterative schemes providerelatively similar convergence structures on the complex plane. However, thereare some numerical methods for which the corresponding basins of attraction areextremely complicated with highly fractal basin boundaries. Moreover, it isproved that the efficiency strongly varies between the numerical methods.

Comparing the basins of attraction for several methods in the circular Sitnikov problem with spheroid primaries