Stochastic analog to phase transitions in chaotic coupled map lattices.
Stochastic dynamical systems are shown to exhibit the same order-disorder phase transitions that have been found in chaotic map lattices. Phase diagrams are obtained for diffusively coupled two-dimensional (2D) lattices, using two stochastic maps and a chaotic one, for both square and triangular geometries, with simultaneous updating. We show how the use of triangular geometry reduces (or even eliminates) the reentrant behavior found in the phase diagrams for the square geometry. This is attributed to the elimination (via frustration) of the antiferromagnetic clusters common to simultaneous updating 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 stochastic map and the chaotic one, evaluated in an early work by Marcq et al. [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 lattices, and is a basic ingredient for models that belong to this weak Ising universality. However, the fact that the second stochastic map, which has an extremely simple local dynamics, seems to fall in the 2D Ising universality class, suggests that some minimal local complexity is also needed to generate the specific correlations that end up giving non-Ising critical behavior.
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Departamento de Física Aplicada, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Unidad Mérida, Apartado Postal 73 Cordemex, 97310 Mérida, Yucatán, Mexico. sastre@kin.cieamer@conacyt.mx
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