This work aims to study the complex dynamics of rational maps, which, according to the Sullivan component, include sets of forward cyclic iterations, which include both equivalent and doubly connected components, where according to study the family of complex rational maps given by   Qa (z) = 2an+1 z–n + z2n – a3n+1/(zn (z2n – an–1)  where n ≥ 2 and a ∈ Λ ≔ C\{0} – {a : an+1 = 1 or a2n+2 = 1} So, the family would be considered a perturbation of a polynomial map. Therefore, we get the results. So Qa has neither infinitely connected attracting components nor equivalent (equivalent) components if the orbits of all free critical points are not attracted to the origin or infinity. Also, Qa  does not contain Herman’s ring. So Qa has no Siegel disc.