that in Fig. 2 of Mikkola et al. (2004). Now the object, after leaving the L$_5$ Venus Trojan location, quickly enters the horseshoe-like phase. It does not inmediately become an $L_{4}$ Trojan like in Model 1. Consistently, Fig. \[orbele2\] shows differences with respect to Fig. \[orbele1\] although they exhibit similar behaviour in the time interval -7,000 to +1,000 yr from now. During the horseshoe-like phase, the semi-major axis remains oscillating (but with larger amplitude) around the value of the semi-major axis of Venus.

The inclusion of the Moon in the calculations has a significant impact on the evolution of the distances in Fig. \[distance2\]. Now encounters as close as 0.0025 AU (not shown) are possible for both Mercury and the Earth so we may say that the resonant behaviour of 2002 VE$_{68}$ is controlled by both the Earth and Mercury. On the other hand, relatively close encounters with Venus are only observed prior to the quasi-satellite phase (multiple encounters under 0.03 AU) and after leaving the quasi-satellite dynamical state (0.013 AU).

Model 3
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The JPL Small-Body Database indicates that the three largest asteroids, (1) Ceres, (2) Pallas and (4) Vesta, are minor perturbers of 2002 VE$_{68}$ and in our 