We use numerical integrations to investigate the dynamical evolution of
resonant Trojan and quasi-satellite companions during the late stages of
migration of the giant planets Jupiter, Saturn, Uranus, and Neptune. Our
migration simulations begin with Jupiter and Saturn on orbits already well
separated from their mutual 2:1 mean-motion resonance. Neptune and Uranus are
decoupled from each other and have orbital eccentricities damped to near their
current values. From this point we adopt a planet migration model in which the
migration speed decreases exponentially with a characteristic timescale tau
(the e-folding time). We perform a series of numerical simulations, each
involving the migrating giant planets plus test particle Trojans and
quasi-satellites. We find that the libration frequencies of Trojans are similar
to those of quasi-satellites. This similarity enables a dynamical exchange of
objects back and forth between the Trojan and quasi-satellite resonances during
planetary migration. Furthermore, under the influence of these secondary
resonances quasi-satellites can have their libration amplitudes enlarged until
they undergo a close-encounter with their host planet and escape from the
resonance. High-resolution simulations of this escape process reveal that ~80%
of Jovian quasi-satellites experience one or more close-encounters within
Jupiter's Hill radius (R_H) as they are forced out of the quasi-satellite
resonance. As many as ~20% come within R_H/4 and ~2.5% come within R_H/10.
Close-encounters of escaping quasi-satellites occur near or even below the
2-body escape velocity from the host planet