AbstractSmall amplitude variable red giants were discovered long time ago (Eggen 1969, Henry et al. 2000, Jorison et al. 1997). The revolutionary turning point for the study of this type of variable was due to the work of Wood (2000) and Soszyński et al. (2004) based on the very large stellar sample of the Magellanic clouds MACHO project and OGLE photometric data. By using our non-local time-dependent theory of convection (Xiong 1989), we carried out a linear stability survey for red giants with initial mass of M = 1 − 3M_⊙. Explanations for the observed variabilities and analysis of mode identifications in red giant stars are presented in this work.