We study a piecewise linear version of a one-component, one-dimensional reaction-diffusion bistable model, with the aim of analyzing the effect of boundary conditions on the formation and stability of stationary patterns. The analysis proceeds through the study of the behavior of the Lyapunov functional in terms of a control parameter: the reAectivity at the boundary. We show that, in this example, this functional has a very simple and direct geometrical interpretation.

The model introduced by Van den Broeck, Parrondo and Toral [Phys. Rev. Lett. 73, 3395 (1994)]-leading to a second-order-like noise-induced nonequilibrium phase transition which shows reentrance as a function of the (multiplicative) noise intensity σ-is investigated beyond the white-noise assumption. Through a Markovian approximation and within a mean-field treatment it is found that-in striking contrast with the usual behavior for equilibrium phase transitions-for noise self-correlation time τ > 0, the stable phase for (diffusive) spatial coupling D → ∞ is always the disordered one. Another surprising result is that a large noise "memory" also tends to destroy order. These results are supported by numerical simulations.

We address a recently introduced model describing a system of periodically coupled nonlinear phase oscillators submitted to multiplicative white noises, wherein a ratchetlike transport mechanism arises through a symmetry-breaking noise-induced nonequilibrium phase transition. Numerical simulations of this system reveal amazing novel features such as negative zero-bias conductance and anomalous hysteresis, explained by performing a strong-coupling analysis in the thermodynamic limit. Using an explicit mean-field approximation, we explore the whole ordered phase finding a transition from anomalous to normal hysteresis inside this phase, estimating its locus, and identifying (within this scheme) a mechanism whereby it takes place.