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

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.

Figure 1

(Color online) Trajectory $x_n$ of the logistic map with parameter $r=4$ (lower panel). Initial value $x_0=0.857$. Square symbols stand for the analytic solution Eq. (2) or, equivalently, the MP numerical computation of the logistic map. Circles (red) represent DP computation of Eq. (1). Triangles (blue) represent DP computation of Eq. (3). With the same symbols code, the upper panel shows in semilogarithmic scales the absolute error of both DP computations for the 100 first iterations.

Figure 2

(Color online) Histograms for the invariant density of the logistic map (1) with parameter $r=4$. Black solid (red open) symbols stand for MP (DP) computations. The blue solid line is the function $300\pi \rho \left(x\right)$. A DP computation with Eq. (3) yields a similar histogram.

Figure 3

Snapshots of a chaotic trajectory up to $n=360$ at the edge of the period-3 window: $r=1+\sqrt{8}-{10}^{-3}$ exhibiting intermittency. Bottom (top) subpanels correspond to DP (MP) computations. The initial point is the same for both computations, $x_0=0.25$. Vertical scales (nonlabeled) stand for the interval [0,1].

Figure 4

(Color online) DP and MP histograms of invariant densities at the edge of period 3. Top panel is for $r=1+\sqrt{8}+{10}^{-3}$, regular stable regime. Bottom panel is for $r=1+\sqrt{8}-{10}^{-3}$, chaotic regime: only the color online version allows a tiny difference between DP (red line) and MP (black line) to be observed. Arrows bound the largest error $\Delta$ found, defined in Sec. 4, which holds for both panels.

Figure 5

(Color online) DP and MP histograms of invariant densities at the edge of period 6. Top panel is for $r=3.627$, regular regime. Bottom panel is for $r=3.626$, chaotic regime: only the color online version allows one to notice a tiny difference between DP (red line) and MP (black line). Arrows delimit the largest error $\Delta$, defined in Sec. 4. This value holds also for the upper panel.

Figure 6

(Color online) Error statistics of DP trajectories of the logistic map, Eq. (1), with parameter $r=4$ are represented by circles. The solid straight line (blue) stands for the analytic formula, Eq. (8). Computations from Eq. (3) yield similar results and are not given for the sake of clarity.

Figure 7

(Color online) Histograms of the error [see Eq. (6)] statistics of DP trajectories of the logistic map. Results for several parameter values are shown. Curves are explicitly labeled with the value of $r$ for the sake of clarity. The inset shows the case $r=3.6$ which is off scale.

Figure 8

(a) Largest error $\Delta$ in the DP precision computation of the logistic map as a function of $R\equiv r-r_{\infty }$. The inset corresponds to the same curve in log-log scales. (b) Bifurcation diagram for the logistic map. Abscissas start at $r=r_{\infty }$, which establishes the proper axes correspondence with (a) and allows one to observe that the discontinuities in $\Delta \left(r\right)$ correlate with the merging of segments of $\rho \left(x\right)$.

Figure 9

(Color online) Semilogarithmic plot for relative location $R_n$ of jumps in $\Delta \left(r\right)$ and height of discontinuities $D_n$. Linear fits are suggested by Eqs. (9, 10).

Figure 10

(Color online) Leading digit statistics from the logistic map errors (squares) and Benford’s law (circles). Solid symbols are referred to linear scales (bottom and left axes). Empty symbols are referred to ${\log }_{10}\text{−}{\log }_{10}$ scales (top and right axes).