  The study investigates detonations with multiple quasi-steady velocities that
have been observed in the past in systems with multi-peaked thermicity, using
Fickett's detonation analogue. A steady state analysis of the travelling wave
predicts multiple states, however, all but the one with the highest velocity
develop a singularity after the sonic point. Simulations show singularities are
associated with a shock wave which overtakes all sonic points, establishing a
detonation travelling at the highest of the predicted velocities.
  Under a certain parameter range, the steady-state detonation can have
multiple sonic points and solutions. Embedded shocks can exist behind sonic
points, where they link the weak and strong solutions. Sonic points whose
characteristics do not diverge are found to be unstable, and to be the source
of the embedded shocks. Numerical simulations show that these shocks are only
quasi-stable. This is believed to be due in part to a feature of the model
which permits shocks anywhere behind a sonic point.
