  We investigate the gravonuclear instabilities reported by Bono et al.
(1997a,b) during the onset of helium-shell burning at the end of the
horizontal-branch (HB) phase. These instabilities are characterized by
relaxation oscillations within the helium shell which lead to loops in the
evolutionary tracks. We find the occurrence of these instabilities depends
critically on how the breathing pulses are suppressed near the end of the HB
phase. If they are suppressed by omitting the gravitational energy term in the
stellar structure equations, then the helium profile within the core at the end
of the HB phase will contain a broad region of varying helium abundance. The
helium-burning shell which forms in this region is too thick to be unstable,
and gravonuclear instabilities do not occur. If, on the other hand, the
breathing pulses are suppressed by prohibiting any increase in the central
helium abundance, then the final helium profile can exhibit a large
discontinuity at the edge of the helium-exhausted core. The helium shell which
forms just exterior to this discontinuity is then much thinner and can be
thermally unstable. Even in this case, however, the gravonuclear instabilities
disappear as soon as the nuclear burning broadens the helium shell into its
characteristic S-shape. We conclude that the gravonuclear instabilities found
by Bono et al. are a consequence of the ad hoc procedure used to suppress the
breathing pulses.
