  We present a theoretical study of an exciton-polariton annular microcavity
with an additional quasiperiodic structure along the ring which is implemented
in the form of a bicosine dependence. We demonstrate that for a sufficiently
strong quasiperiodic modulation, the microcavity features a sharp mobility edge
separating a cluster of localized states from the rest of the spectrum
consisting of states extended over the whole ring. Localized modes can be
excited using a resonant pump whose topological charge determines the phase
distribution of excited patterns. Repulsive polariton interactions make the
resonance peaks distinctively asymmetric and enable the formation of
multistable states which feature the attractor-like dynamical behavior \rev{and
hysteresis}. We also demonstrate that the localized states can be realized in a
biannular cavity that consists of two rings, each having periodic modulation,
such that the periods of two modulations are different.
