The pulsational evolution of stars along the AGB

The pulsational evolution of stars along the asymptotic giant branch (AGB) through the helium shell flashes (HSFs) is described. This evolution is found to be very different from the commonly assumed one, which makes use of the well known core mass - luminosity relation (CMLR). The essential differences between the two evolutionary scenarios are the temporal changes of the luminosity and period, which may differ by up to an order of magnitude, and the lack of monotony in the present scenario allowing for the possibility to observe highly evolved cores in the low luminosity - short period region. Possible observational implications are discussed.