Double precision errors in the logistic map: Statistical study and dynamical interpretation

Abstract

The nature of the round-off errors that occur in the usual double precision computation of the logistic map is studied in detail. Different iterative regimes from the whole panoply of behaviors exhibited in the bifurcation diagram are examined, histograms of errors in trajectories given, and for the case of fully developed chaos an explicit formula is found. It is shown that the statistics of the largest double precision error as a function of the map parameter is characterized by jumps whose location is determined by certain boundary crossings in the bifurcation diagram. Both jumps and locations seem to present geometric convergence characterized by the two first Feigenbaum constants. Eventually, a comparison with Benford's law for the distribution of the leading digit of compilation of numbers is discussed.