  A finite-dimensional Markovian open quantum system will undergo quantum jumps
between pure states, if we can monitor the bath to which it is coupled with
sufficient precision. In general these jumps, plus the between-jump evolution,
create a trajectory which passes through infinitely many different pure states,
even for ergodic systems. However, as shown recently by us [Phys. Rev. Lett.
\textbf{106}, 020406 (2011)], it is possible to construct {\em adaptive}
monitorings which restrict the system to jumping between a finite number of
states. That is, it is possible to track the system using a {\em finite state
machine} as the apparatus. In this paper we consider the question of the
stability of these monitoring schemes. Restricting to cyclic jumps for a qubit,
we give a strong analytical argument that these schemes are always stable, and
supporting analytical and numerical evidence for the example of resonance
fluorescence. This example also enables us to explore a range of behaviors in
the evolution of individual trajectories, for several different monitoring
schemes.
