Static parametric fluctuations give nonstatistical behavior in uncoupled chaotic systems

Abstract

We show that an ensemble of logistic maps, with parameters distributed in some range inside a chaotic region, cannot be statistical. This is so because any parameter range for this map includes periodic windows. The effects of this periodicity in the averages of the system can be estimated, and are also apparent in the power spectra of average values. As a counterexample, we show that the behavior for tent maps, where chaotic regions do not include periodic windows, is statistical.