Cyclic contractions generalize the usual contractivities in metric spaces and $b$-MSs. In this paper, we enhance several fixed point theorems related to cyclic (i) Banach self-maps, (ii) Chatterjea contractivities, (iii) Kannan self-mappings, (iv) Ćirić and Hardy-Rogers, and (v) Reich contractions including local versions in $b$-metric spaces while also delineating the associated dynamics. Especially noteworthy is the expansion of the results concerning both fixed and periodic points, which are substantiated across a wider spectrum of ratio values for the aforementioned cyclic contractions within this class of spaces. Additionally, the convergence of Picard iterations towards the fixed point is rigorously established.