  We describe a new mechanism for pulsations in evolved stars: relaxation
oscillations driven by a coupling between the luminosity-dependent mass-loss
rate and the H fuel abundance in a nuclear-burning shell. When mass loss is
included, the outward flow of matter can modulate the flow of fuel into the
shell when the stellar luminosity is close to the Eddington luminosity $L_{\rm
Edd}$. When the luminosity drops below $L_{\rm Edd}$, the mass outflow declines
and the shell is re-supplied with fuel. This process can be repetitive. We
demonstrate the existence of such oscillations and discuss the dependence of
the results on the stellar parameters. In particular, we show that the
oscillation period scales specifically with the mass of the H-burning
relaxation shell (HBRS), defined as the part of the H-burning shell above the
minimum radius at which the luminosity from below first exceeds the Eddington
threshold at the onset of the mass loss phase. For a stellar mass $M_*\sim
0.7\Msun$, luminosity $L_*\sim 10^4\Lsun$, and mass loss rate $|\dot M|\sim
10^{-5}\Msun$ yr$^{-1}$, the oscillations have a recurrence time $\sim 1400$
years $\sim 57\tau_{\rm fsm}$, where $\tau_{\rm fsm}$ is the timescale for
modulation of the fuel supply in the HBRS by the varying mass-loss rate. This
period agrees with the $\sim$ 1400-year period inferred for the spacings
between the shells surrounding some planetary nebulae, and the the predictied
shell thickness, of order 0.4 times the spacing, also agrees reasonably well.
