  We consider the superstable cycles of the Q-state Potts (QSP) and the
three-site interaction antiferromagnetic Ising (TSAI) models on recursive
lattices. The rational mappings describing the models' statistical properties
are obtained via the recurrence relation technique. We provide analytical
solutions for the superstable cycles of the second order for both models. A
particular attention is devoted to the period three window. Here we present an
exact result for the third order superstable orbit for the QSP and a numerical
solution for the TSAI model. Additionally, we point out a non-trivial
connection between bifurcations and superstability: in some regions of
parameters a superstable cycle is not followed by a doubling bifurcation.
Furthermore, we use symbolic dynamics to understand the changes taking place at
points of superstability and to distinguish areas between two consecutive
superstable orbits.
