The new nonstationary model problem is considered. Its solution generalizes by form the known particular Mestschersky-Vinti solution in a two-body problem of variable mass. The equations of the corresponding perturbed motion are deduced. In the case of a two-body problem of variable mass μ. the perturbing force is proportional to second temporal derivative from the value μ-1 . It is possible to describe with a good approximation such qualitative effects in this problem as a trapping and disintegration on a basis of properties of the model problem. Let us consider the example of a trapping.

Previous studies of close encounters of minor bodies with Jupiter have shown that the perturbations are stronger either if the encounter is very deep or if the velocity of the minor body relative to the planet is low. In the present research we investigate the effects of low velocity encounters between fictitious minor bodies and the four outer planets. Two possible outcomes of this type of encounter are the temporary satellite capture of the minor body by the planet, and the exchange of perihelion with aphelion of the minor body orbit. Different occurrence rates of these processes are found for different planets, and the implications for the orbital evolution of minor bodies in the outer Solar System are discussed.

We study numerically the competition between the Poynting-Robertson drag and the gravitational interaction of grains with Jupiter near orbital resonances. The computations are based on the plane elliptic restricted three body problem. Numerical investigations show that the grains always cross the resonance region without any oscillation, except in the special case where the grains were initially inside the resonance. Such grains are temporarily trapped, then due to the drag they are ejected out of the resonance. The trapping time of a particle turns out to be much more important in the 3/2 and 2/1 commensurabilities than in the others.

A numerical exploration of numerous orbits for different initial conditions and different sizes of grains has been performed. The trapping time appears to be closely connected to the size of the librator-type orbits regions; it increases with the initial eccentricity of the orbit, and is also proportional to the radius and the density of the particle.

The problem of constructing the stochasticity criterion of a degenerate near-integrable dynamical system is considered. We show that in certain cases the above criterion can be found as the limit of the corresponding stochasticity criteria of a family of non-degenerate systems, whose limit is the degenerate one.