Superstable cycles for antiferromagnetic Q-state Potts and three-site interaction Ising models on recursive lattices

•Superstability in the QQ-state Potts and antiferromagnetic Ising models on recursive lattices is analyzed.•Some analytic expressions are obtained for the superstable cycles of the second and the third orders.•A non-trivial connection between bifurcation and superstability is established in the period three window.•The symbolic dynamics technique is implemented for tracking the changes taking place at points of superstability.

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.