The dynamical behaviour of the envelopes of four asymptotic-giant-branch stars of composition (X, Z)=(0.68, 0.02) and luminosity log L/L⊙ = 3.41, 3.60, 3.85 and 4.14 has been studied. At log L/L⊙ = 3.41, the envelope pulsates in the first-overtone mode and exhibits properties similar to those of the Mira variables. The more luminous models were all found to pulsate in the fundamental mode while simultaneously undergoing violent envelope relaxation oscillations. A significant amount of mass loss occurred from the model with luminosity log L/L⊙ = 3.60. It is suggested that a redgiant star undergoing envelope relaxation oscillations would be recognised as a symbiotic star. In the two most luminous models, a distinct outward-moving shell containing almost all the hydrogen-rich material in the star is formed. A suggestion is made that a further increase in luminosity would completely eject this shell to form a planetary nebula.