Nonlinear pulsations of stars with initial mass 3 M_\odot on the   asymptotic giant branch
  Pulsation period changes in Mira type variables are investigated using the stellar evolution and nonlinear stellar pulsation calculations. We considered the evolutionary sequence of stellar models with initial mass 3 Msol and population I composition. Pulsations of stars in the early stage of the asymptotic giant branch are shown to be due to instability of the fundamental mode. In the later stage of evolution when the helium shell source becomes thermally unstable the stellar oscillations arise in either the fundamental mode (for the stellar luminosity L < 5.4e3 Lsol) or the first overtone (L > 7e3 Lsol). Excitation of pulsations is due to the kappa-mechanism in the hydrogen ionization zone. Stars with intermediate luminosities 5.4e3 < L/Lsol < 7e3 were found to be stable against radial oscillations. The pulsation period was determined as a function of evolutionary time and period change rates \dot\Pi were evaluated for the first ten helium flashes. The period change rate becomes the largest in absolute value (\dot\Pi/\Pi\approx -0.01 per year) between the helium flash and the maximum of the stellar luminosity. Period changes with rate |\dot\Pi/\Pi| > 1e-3 per year take place during approx 500 years, that is nearly one hundredth of the interval between helium flashes. 
Yuri Fadeyev
astro-ph.SR