Logarithmic Scaling in the Stationary-Nonstationary Chaos Transition

Statistical features of the transition process from stationary to nonstationary chaos are studied using modified Bernoulli maps. The stationary-nonstationary chaos transition process can be generated by varying the value of the parameter that controls the intensity of the intermittency. The measure-theoretical structures of the chaos transition process are significantly different from the time-independent case, and new statistical phenomena appear, even when the value of the system parameter is changed continuously in time. The temporal behavior of the transition process is generally separated into three characteristic phases. The first phase appears in the initial stage of the transition, and the third one in the last stage, where the transition is almost finished, but the most interesting phase is the second one, which continues for long period. The entire transition process is analyzed using the renewal function, which describes the temporal behavior of the mean accumulated number of intermittent jumps. The logarithmic scaling relation appearing in the second phase is studied in detail with finite-range statistics. Finally, the statistical laws of the stationary-nonstationary chaos transition are discussed from the viewpoint of the modeling of seismological phenomena, and it is shown that seismological data are accounted for quite well by a metaphor model in terms of the stationary-nonstationary chaos transition.