Migration of asteroids from the 3/1 and 5/2 resonances with Jupiter to   the Earth
  The orbital evolution of asteroids from the 3:1 and 5:2 resonances with Jupiter under the gravitational influence of planets was investigated both by the Bulirsh-Stoer method and by a symplectic method. The probabilities of collisions of asteroids with the terrestrial planets were calculated for orbital elements obtained with a step of 500 yr. The ratio of asteroids from the 3:1 resonance that were ejected into hyperbolic orbits to those collided the Sun was by an order of magnitude larger for direct integrations than for a symplectic method. The probability of a collision of an asteroid with the Earth for the 3:1 resonance was by a factor of 4 larger than that for the 5:2 resonance. The ratio of the probability of a collision of a body with a planet to the mass of the planet for Mars was by a factor of 2 larger than that for Earth and Venus. 
S.I. Ipatov