  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 (~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.
