Should "stable"/"unstable" actually be "periodic"/"aperiodic"?

[duetosymmetry]

In this context, "stable" would probably be taken to mean dynamical stability, i.e. if you start with initial conditions which are slightly perturbed, do the two sets of solutions diverge exponentially in time? I am not aware of stability studies of extremely-finely-tuned periodic orbits, but I would be extremely surprised if they were stable.