The orbital evolution of about 20000 Jupiter-crossing objects and 1500 resonant asteroids under the gravitational influence of planets was investigated .
The rate of their collisions with the terrestrial planets was estimated by computing the probabilities of collisions based on random-phase approximations and the orbital elements sampled with a 500 yr step .
The Bulirsh-Stoer and a symplectic orbit integrators gave similar results for orbital evolution , but sometimes gave different collision probabilities with the Sun .
For orbits close to that of Comet 2P , the mean collision probabilities of Jupiter-crossing objects with the terrestrial planets were greater by two orders of magnitude than for some other comets .
For initial orbital elements close to those of Comets 2P , 10P , 44P and 113P , a few objects ( \sim 0.1 % ) got Earth-crossing orbits with semi-major axes a < 2 AU and moved in such orbits for more than 1 Myr ( up to tens or even hundreds of Myrs ) .
Some of them even got inner-Earth orbits ( i.e. , with aphelion distance Q < 0.983 AU ) and Aten orbits .
Most former trans-Neptunian objects that have typical near-Earth object orbits moved in such orbits for millions of years ( if they did not disintegrate into mini-comets ) , so during most of this time they were extinct comets .