The Chaotic Properties of the Q-state Potts Model on the Bethe Lattice:Q<2

The Q- state Potts model on the Bethe lattice is investigated for Q<2. The magnetization of this model exhibits a complicated behavior including both the period doubling bifurcation and chaos. The Lyapunov exponents of the Potts-Bethe map are considered as order parameters. We find a scaling behavior in the distribution of Lyapunov exponents in fully developed chaotic case. Using the thermodynamic formalism of dynamical systems have we investigated the nonanalytic behavior in the distribution of Lyapunov exponents and located the point of phase transition of the ‘‘chaotic free energy’’.