tokens: the stable nature of the inclination oscillations, the incompleteness of detecting high-eccentricity and high-inclination objects, and the greater likelihood of low-inclination objects impacting the Earth (as mentioned in Section 1 ) suggest that extensive observation of high-inclination locations of vertically unstable quasi-satellite orbits may be helpful in detecting populations of potentially hazardous asteroids in advance before they reduce inclination. The long-term stable inclination oscillations associated with quasi-satellite orbits obtained in the simplified model in this study may appear to be contradictory to the shorter lifetime of , of only tens of thousands years, on its quasi-satellite motion. We do not expect long lifetimes of such objects in the real world experiencing close approaches to planets at low-inclination states (Wajer Królikowska 2012 ), but the stable nature of the inclination oscillations in the simplified model implies an ability to capture objects even in the real world. Certainly, was trapped in the quasi-satellite motion exhibiting inclination oscillations from a chaotic orbit (see fig. 5 in Kinoshita Nakai 2007 ). The stable nature is significant in our conjecture, not because of the longevity after experiencing low-inclination states, but because of the ability to capture objects in the inclination oscillations at high inclinations before reducing their inclinations. This paper focuses on the fundamental dynamics of vertical instability and inclination excitations of co-orbital orbits in the circular restricted three-body problem, but limitations exist due to the simplified assumptions of the model and additions of various perturbations, such as the eccentricity of Jupiter, and the gravitational effects of other planets could be promising directions. In such non-autonomous systems, computations of invariant manifolds associated with resonant periodic orbits (Voyatzis Antoniadou 2018 ) and quasi-periodic invariant tori (Olikara Scheeres 2012 ) would be natural extensions of the present work. 5 CONCLUSIONS This paper explored the effects of vertical instability on inclination excitations of co-orbital orbits in the Sun--Jupiter circular restricted three-body problem. Computations of invariant manifolds associated with vertical instability of the planar co-orbital periodic orbits enabled quantitative analyses of inclination excitations. Quasi-satellite orbits exhibited long-term stable, highly eccentric, substantial inclination oscillations between near zero and tens of degrees without escaping from quasi-satellite states. We pointed out the possibility that populations of undetected potentially hazardous asteroids exist at high-inclination locations of Jupiter’s vertically unstable quasi-satellite orbits. Analyses of bifurcation and phase-space structures revealed that stable, three-dimensional families of periodic quasi-satellite orbits host inclination oscillations in a stable manner. However, planar
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