The averaged, hyperbolic, restricted three-body problem

The plane, singly averaged, hyperbolic, restricted three-body problem in the case of small eccentricities of the perturbing body is analyzed in the article. The general solution of the linearized system of differential equations in the form of series in powers of the small parameter is found in the case when the first term is taken in the perturbation function. It is shown that stability of the unperturbed motion is retained when the orbital eccentricity of the perturbing body is relatively small and the perturbed body is sufficiently close to the central body. The exact and singly averaged problems are integrated numerically, and the linearized system is calculated on a computer from the equations of the general solution. The analytical and qualitative conclusions are confirmed by comparing the results obtained.