Fix-lines and stability domain in the vicinity of the coupled third order resonance

The single particle stability in a circular accelerator is of concern especially for operational regimes involving beam storage of hours. In the proximity to a resonance this stability domain shrinks, and the phase space fragments into a jungle of exotic objects like for instance "fix-lines". The concept of fix-points is easily understandable in a 2D phase space. It becomes quite challenging when the effect of resonances is considered in the 4D phase space, which leads then to the concept of fix-lines. In this paper we investigate the fix-lines in the proximity of a coupled third order resonance and find the relation of these objects with the stability of motion.