Dynamical evolution of a self-organized-critical percolation model

In this work we analyze a theoretical model showing self-organized criticality. We study the automatic search of the critical point of the site percolation model in the square lattice. The trajectory of the system going to the critical point and oscillating around it is analyzed as a dynamical problem. A simple map that describes the dynamics close to the critical point is constructed. The dynamics has two characteristic time scales: a time for attaining the critical point and a time for loosing correlation. This last point is discussed in connection with brown and white noise observed in the Fourier spectrum. Despite a short time highly correlated behavior, the system shows an ergodic trend for long time scales.