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

TL;DR

This paper analyzes superstable cycles in Q-state Potts and three-site interaction Ising models on recursive lattices, providing analytical and numerical solutions, and exploring bifurcation-superstability relationships using symbolic dynamics.

Contribution

It offers the first analytical solutions for second order superstable cycles and exact third order superstable orbit results, linking bifurcations and superstability.

Findings

Analytical solutions for second order superstable cycles.

Exact third order superstable orbit for QSP.

Numerical third order superstable orbit for TSAI.

Abstract

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.