A number of Jupiter comets such as *Oterma* and *Gehrels 3*
make a rapid transition from heliocentric orbits outside the orbit of
Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa.
During this
transition, the comet is frequently captured temporarily by Jupiter for
one to several orbits around Jupiter. The interior heliocentric orbit
is typically close to the 3:2 resonance (three revolutions
around the Sun in two Jupiter periods) while the exterior heliocentric
orbit is near the 2:3 resonance (two revolutions around the Sun in
three Jupiter periods).

An important feature of the dynamics of these comets is that during the
transition, the orbit passes close to the libration points L1 and L2.
The points L1 and L2 are two of the five
equilibrium points for the restricted three-body problem for the
Sun-Jupiter system. Amongst
the equilibrium points, L1 and L2 are the ones closest to
Jupiter, lying on either side of Jupiter along the Sun-Jupiter line.

We conclude that studying the libration point invariant manifold
structures for L1 and L2 is a starting point for understanding
the capture and resonance transition of these comets.
The recently discovered heteroclinic connection between pairs of
unstable periodic orbits (one around the L1 and the
other around L2) has significant
implications for the aforementioned resonance transition and
temporary capture of Jupiter comets.

The stable and unstable invariant manifold "tubes" associated
to libration point periodic orbits, of which the
heteroclinic connections are a part, act as conduits in the phase space
transporting material to and from Jupiter and between the interior and
exterior of a Jupiter's orbit.
Temporary capture and collision orbits such as comet
*Shoemaker-Levy 9*'s can be modeled via these
dynamical channels.

This is joint work with
W. Koon,
M. Lo, and
J. Marsden.