Clustering of extreme and recurrent events in deterministic chaotic systems

We study the nontrivial clustering properties of extreme or recurrent events in the context of deterministic chaotic systems. We find that correlations between return times of such events can depend nonmonotonically on the threshold used to define the events, which leads to counterintuitive behavior. In particular, the distribution of the conditional return intervals can indicate clustering as well as repelling of extreme events for the same condition but different thresholds—in sharp contrast to what has been observed for stochastic processes with long-range correlations as well as for independent and identically distributed random variables. This has important implications for the time-dependent hazard assessment of extreme events, indicating that possible threshold dependencies should always be taken into account.

Figure 1

Autocorrelation coefficient of the sequence of return intervals of extreme events as a function of threshold values $x_{\text{th}}$ for different lags $l$. The curves belonging to the different lags are marked by different symbols [$l=1$: (red) squares; $l=2$: (green) circles; $l=3$: (blue) reverse triangles; $l=4$: (cyan) diamonds]. Nonmonotonic dependence on both threshold and lag can be observed. The vertical lines represent some of the threshold values for which all coefficients must be zero (see text for a discussion).

Figure 2

Autocorrelation function of the sequence of return intervals of extreme events for two different threshold values representing clustering and repelling of return intervals, respectively.

Figure 3

The figure shows the conditional probability $P_{x_{\text{th}}}(r|r_0)$ divided by the unconditional probability $P_{x_{\text{th}}}(r)$ as a function of $r/{\mathrm{r̄}}_{x_{\text{th}}}$ (return interval in units of the mean return interval). For all data sets we chose $r_0/{\mathrm{r̄}}_{x_{\text{th}}}=\sqrt{2}$. Data for the threshold values $x_{\text{th}}=0.293$ (triangles), $x_{\text{th}}=0.529$ (squares), $x_{\text{th}}=0.646$ (diamonds), $x_{\text{th}}=0.717$ (reverse triangles), and $x_{\text{th}}=0.764$ (circles) are presented. The results have been obtained numerically from a data set with ${10}^9$ return intervals per threshold value.

Figure 4

The figure shows the conditional mean return interval ${\mathrm{r̄}}_{x_{\text{th}}}(r_0)$ in units of the unconditional mean return interval ${\mathrm{r̄}}_{x_{\text{th}}}$ vs the condition $r_0/{\mathrm{r̄}}_{x_{\text{th}}}$. The same threshold values as in Fig. 3 have been used. The results have been obtained numerically from a data set with ${10}^9$ return intervals per threshold value.

Figure 5

Like Fig. 3 but for the cusp map and $r_0/{\mathrm{r̄}}_{x_{\text{th}}}=0.6$. Data for the threshold values $x_{\text{th}}=-0.0954$ (triangles), $x_{\text{th}}=0.1056$ (squares), $x_{\text{th}}=0.2254$ (diamonds), and $x_{\text{th}}=0.5101$ (reverse triangles) are presented. The results have been obtained numerically from a data set with $2\times {10}^9$ return intervals per threshold value.

Figure 6

Like Fig. 4 but for the cusp map. The same threshold values as in Fig. 5 have been used. The results have been obtained numerically from a data set with $2\times {10}^9$ return intervals per threshold value.