Failure of a mean-field approach for the Miller-Muse phase transition

The diffusively coupled lattice of odd-symmetric chaotic maps introduced by Miller and Huse undergoes a continuous ordering phase transition, belongin g to a universality class close but not identical to that of the two-dimens ional Ising model. Here we consider a natural mean-field approach for this model, and find that it does not have a well-defined phase transition. We s how how this is due to the coexistence of two attractors in its mean-field description, for the region of interest in the coupling. The behavior of th e model in this limit then becomes dependent on initial conditions, as can be seen in direct simulations.