Extreme events due to localization of energy

We study a one-dimensional chain of harmonically coupled units in an asymmetric anharmonic soft potential. Due to nonlinear localization of energy, this system exhibits extreme events in the sense that individual elements of the chain show very large excitations. A detailed statistical analysis of extremes in this system reveals some unexpected properties, e.g., a pronounced pattern in the interevent interval statistics. We relate these statistical properties to underlying system dynamics and notice that often when extreme events occur the system dynamics adopts (at least locally) an oscillatory behavior, resulting in, for example, a quick succession of such events. The model therefore might serve as a paradigmatic model for the study of the interplay of nonlinearity, energy transport, and extreme events.

Figure 1

Heat maps showing the evolution of the amplitudes $q_n\left(t\right)$ for 100 coupled units, as a function of time, for two different values of the coupling strength: $\kappa =0.1$ (top) and $\kappa =0.4$ (bottom). All amplitudes $q_n\left(t\right)\ge 4$ are in blue (black). Here $t$ denotes time and $n$ the location of the units on the chain. The oscillating island structures are clearly visible.

Figure 2

Number of $q_{\mathrm{thres}}$ (filled circles) and $E_{\mathrm{thres}}$ (crosses) threshold crossings as a function of the coupling strength $\kappa$.

Figure 3

The curves at the top of the figure show the evolution of the localization measure $L_N$, the middle curves are the potential energy, and the bottom curves are the kinetic energy. For each coupling strength (see key) the curves have been obtained using the same ensemble of 100 trajectories. Note that there is a break in the axis at $t=45$.

Figure 4

The average energetic excitation pattern in the vicinity of the location of the maximum $i_m$ when the excitation exceeds the $E_{\mathrm{thres}}=3$ threshold. The logarithmic representation (inset) shows that it is, to a good approximation, the superposition of some constant background and a symmetric exponential part, hence we observe exponential localization in space.

Figure 5

Threshold crossing statistical distributions for $q$ (left) and $En$ (right). Row 1, maximum amplitudes for each event considered extreme; 2, event duration times for events considered extreme; 3, waiting times between events happening anywhere on the chain; 4, waiting times between consecutive events happening at the same chain location. More details in the text.

Figure 6

Lin-log plots for some of the results contained in Fig. 5; panels (1), (2), (3), and (4) correspond, respectively, to panels (a), (e), (d), and (h) from Fig. 5 (empty symbols, further details are contained in the key). Also shown are fitted curves obtained using the function $f\left(q\right)=aexp(-b{q}^c)$ (filled symbols). The corresponding values $b$ and $c$ are contained in the key ($c=1$ in the cases where $c$ is not explicitly given in the key), while $a$ is a constant representing the bin height at $q=4,E=3$. The coupling strength related to each symbol is contained in the key in the top right panel.

Figure 7

Left: Time series for a single chain unit, which crosses the threshold $q_n=4$ multiple times. Right: The stacked histogram shows the distribution of waiting times between consecutive extreme events when decomposed into the number of subthreshold oscillations between events. The red curve (solid line) shows the full distribution (from previous section) when no such decomposition has taken place. This curve has been normalized such that its largest peak coincides with that of the stacked histogram. In both panels $\kappa =0.2$.

Figure 8

Spatial autocorrelation function for amplitudes $q$ (left) and energy $E$ (right) with different chain lengths and with coupling strength $\kappa =0.1$.