One-dimensional mapping, modulated phases and Lyapunov exponent for the antiferromagnetic -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 1 2 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.