After the DART impact against Dimorphos, the secondary asteroid in the Didymos system, a spray of fragments will leave the impact site and populate the space around the binary. The particles will evolve under the influence of the gravitational field of the binary asteroid, radiation pressure and the sun tidal force. Using either a polyhedral model or the analytical MacCullagh’s formula for the gravitational attraction of the two massive bodies on the fragments we show that their orbits, with respect to the baricenter of the system, are highly chaotic. Repeated close encounters with both Didymos and Dimorphos lead to fast changes in the particle trajectories preventing an analytical treatment of their potential long-term stability. By slightly changing the initial ejection velocity or the the ejection position, the lifetime of a particle can vary from a few days to several months. Our main goal is to evaluate for how long a ring of ejecta fragments will survive around the asteroid pair (also in view of the arrival of HERA at the binary system about 4 years after the impact) and to model its morphology dictated by the complex dynamical environment. We also intend to explore how much both these properties are (or are not) related to the impact characteristics.