Fig. 5 extends Figs. 1 and 2 by performing a sensitivity analysis on the log-periodic formula with a second log-periodic harmonic (dashed lines in Figs. 1 and 2), in order to assess the reliability and range of uncertainty of the prediction. Using the fit shown in dashed solid lines in Fig. 2, we have generated 10 realizations of an artificial S&P 500 by adding the GARCH noise (described in the previous caption of Fig. 4) to the dashed solid line. We have then fitted each of these 10 synthetic noisy clones of the S&P 500 by our log-periodic formula. This yields the 10 curves shown here in magenta. This test shows that the log-periodic formula with a second log-periodic harmonic (dashed lines in figures 1 and 2) is also providing stable scenarios: the precise timing of the highs and lows remain robust with respect to the realization of the noise. The real S&P 500 price trajectory is shown as the red wiggly line.