  In co-orbital planetary systems, two or more planets share the same orbit
around their star. Here we test the dynamical stability of co-orbital rings of
planets perturbed by outside forces. We test two setups: i) 'stationary' rings
of planets that, when unperturbed, remain equally-spaced along their orbit; and
ii) horseshoe constellation systems, in which planets are continually
undergoing horseshoe librations with their immediate neighbors. We show that a
single rogue planet crossing the planets' orbit more massive than a few lunar
masses (0.01-0.04 Earth masses) systematically disrupts a co-orbital ring of 6,
9, 18, or 42 Earth-mass planets located at 1 au. Stationary rings are more
resistant to perturbations than horseshoe constellations, yet when perturbed
they can transform into stable horseshoe constellation systems. Given
sufficient time, any co-orbital ring system will be perturbed into either
becoming a horseshoe constellation or complete destabilization.
