Given a catalog of virtual asteroids
we
propagate all the orbits from the initial epoch *t*_{0} until some
target epoch *t*_{1} and record all the close approaches to the
terrestrial planets within a distance of 0.1 AU and taking place
between *t*_{0} and *t*_{1}. Note that the list of such approaches for
one solution *X*_{i} is in general different from the list for a
different, but nearby one, say
.
This arises from the MTP
stretching
[Milani et al. 1999] which separates the
solutions on the MTP at a given encounter by a distance

For example, if
,
then two consecutive solutions with
are
separated by *d*=0.2 AU on the MTP, thus they cannot both pass within
0.1 AU of the Earth at that encounter. This fact dictates the
resolution limit: a close approach with
may not be found during the propagation if
.

The sequence of close approaches to a given planet, such as the Earth,
for a given solution *X*_{i}, is controlled by the complex interplay of
resonances and orbital changes resulting from close approaches, as in
the case of 1999 AN_{10} discussed in [Milani et al. 1999]. It is
possible *a posteriori* to explain the occurrence of a given close
approach for one solution *X*_{i} as the result of a sequence of
resonant and/or non-resonant returns. However, the cascade of returns
is often so complex that it generates several possibilities of close
approach every year; some of these possibilities have such a large
stretching, hence such small probabilities of occurring, that they are
not worth considering.

For this reason we have adopted a procedure which works the other way round: we use the scan of 1201 orbits to detect close approaches, in this way selecting the returns with a significant probability, then by inspecting the sequence of returns of a given solution we can easily identify the resonance mechanism which has allowed the returns to take place.