Stochastic analog to phase transitions in chaotic coupled map lattices - art. no. 016207

Stochastic dynamical systems are shown to exhibit the same order-disorder p hase transitions that have been found in chaotic map lattices. Phase diagra ms are obtained for diffusively coupled two-dimensional (2D) lattices, usin g two stochastic maps and a chaotic one, for both square and triangular geo metries, with simultaneous updating. We show how the use of triangular geom etry reduces (or even eliminates) the reentrant behavior found in the phase diagrams for the square geometry. This is attributed to the elimination (v ia frustration) of the antiferromagnetic clusters common to simultaneous up dating of square lattices. We also evaluate the critical exponents for the stochastic maps in the triangular lattices. The strong similarities in the phase diagrams and the consistency between the critical exponents of one st ochastic map and the chaotic one, evaluated in an early work by Marcq et nl . [Phys. Rev. Lett. 77, 4003 (1996); Phys. Rev. E 55, 2606 (1997)] suggest that the "sign-persistence,'' defined as the probability that the local map keeps the sign of the local variable in one iteration, plays a fundamental role in the presence of continuous phase transitions in coupled map lattic es, and is a basic ingredient for models that belong to this weak Ising uni versality. However, the fact that the second stochastic map, which has an e xtremely simple local dynamics, seems to fall in the 2D Ising universality class, suggests that some minimal local complexity is also needed to genera te the specific correlations that end up giving non-Ising critical behavior .