One-dimensional mapping, modulated phases and Lyapunov exponent for the antiferromagnetic Q-state Potts and multi-site exchange interaction Ising models

We describe the bifurcation structure, period doubling and chaos for the antiferromagnetic Q-state Potts model on the Bethe lattice and three-site interaction Ising model on Husimi one in a magnetic field, by using the recursion relation technique. A chaotic behavior of the magnetic susceptibility for the models is observed at low temperatures. The resulting one-dimensional rational mapping has a positive Lyapunov exponent in the region of the chaotic regime for the antiferromagnetic Q-state Potts (Q<2) and three-site interaction Ising models. We discuss modulated phases for the antiferromagnetic Q-state Potts (Q<2 and Q≥2) and three-site interaction Ising model. At low temperatures the Q-state Potts model (Q≥2) has only one modulated phase with 12 pinching corresponding to the 2-cycle. The Q-state Potts (Q<2) and three-site interaction Ising models have an infinite number of modulated phases with different pinching numbers; we construct the first modulated phase after the first bifurcation point.