  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) \'{C}iri\'c 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.
