Three bodies moving in the closed orbits, exposed to the mutual gravitational interactions only, enter gradually, as a rule, into the gravitational resonances: periods of their rotations become related as the rational fractions.
In long term, this phenomenon may lead to either and for the most part, stabilization of the orbits or, for some period ratios, be the cause of the chaotic motions, also.
Investigation of that phenomenon in this work was based on the, so cold, Newton’s three-body system consisting of one massive and 2 considerably smaller bodies ( , ) turning around it. The same procedure may be used for examination of the correspondent restricted three-body problem ( , 0).
Numerical analysis of the problem was carried out in the following way. After discretizing the time sub-space, the differential equations of motions were replaced by the corresponding difference equations. Their approximate solutions were procured by use of the perturbation technique.
It was revealed that the chaotic motions zones are situated around the : ~ 1:3 and 3:1 resonances. This outcome is in accordance with the Wisdom’s result for the Sun-Jupiter-asteroid triad /9/, as well as with the observational data. Extensions of the chaotic motions zone depend on the mass ratios and .
Neglecting all the other gravitational influences, the adopted model may be applied to the Sun, together with any pair of the heavenly bodies, provided that they do not form the binary system. Also, it is not necessary that the small bodies form the mutually closest pair.