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A pulsating star (pulsating variable star) is a variable star that varies in brightness due to pulses in its energy release, i.e., stellar pulsation. Such a star includes layers that expand and contract in an oscillation rather than settle into hydrostatic equilibrium. Among the specific types of stars that fall within this class:
Many or all these reside in the instability strip of the H-R diagram. Long period variables (LPV), variable stars with periods of on the order of days, or hundreds, or thousands of days (various thresholds are used) are generally pulsating giant stars. Among them are Mira variables (a type of AGB star, type M in the General Catalogue of Variable Stars). The term slow irregular variable is something of a catch-all type for stars of that description that are not (yet) further classified. Some white dwarfs vary including ZZ Ceti variables, DAV stars and DBV stars.
Variable stars that are not pulsating stars include some kinds of binary stars. For example, eclipsing binaries are dimmer for us when one of the companions is behind the other.