The dynamics of artificial asteroids on the Trojan-like orbits around Neptune is investigated in this paper. We describe the dependence of the orbital stability on the initial semimajor axis a and inclination i by constructing a dynamical map on the (a,i)-plane. Rich details are revealed in the dynamical map, especially a unstable gap at i=45 deg is determined and the mechanism triggering chaos in this region is figured out. Our investigation can be used to guide the observations.
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