Effect of Noise on Chaos in a One-Dimensional Map

The mechanism and the characteristics of appearance of order in chaos with Gaussian white noise are investigated in the logistic map. This order (noise-induced order) appears as the constancy of the topological entropy and the decrease of the Kolmogorov entropy. Analysis of n-times iterated map reveals that "noise-induced order" is caused by the increase of the length of the laminar region and the subsequent change of the invariant density, where "local structure" overcomes "global structure." We find that "noise-induced order" is accelerated by an increase of the noise period, and takes place only in chaos near period-five and -under windows.