<p>In this paper, we investigate multi-periodicity in a predator–prey system with a fear effect. Overcoming the difficulties in the calculation of focal values and the irreducible decomposition of the algebraic varieties of focal values under some restrictions of biological sense by using the stratified resultant elimination, we find that the weak focus is of multiplicity at most four. Based on this, we identify conditions for the occurrence of exactly one, two, or three small cycles from Hopf bifurcations by determining the independence of focal values. Moreover, applying the Poincaré–Bendixson theorem, we also explore large cycles that are periodic orbits different from those arising from Hopf bifurcations. Further, we prove the existence of the global attractor and obtain its structure by integrating all results about the system. Our work indicates that there are several ways of coexistence for the predator and prey, characterized by monostability, bistability of node–cycle type, and bistability of cycle–cycle type.</p>