Extreme and superextreme events in a loss-modulated ${\mathrm{CO}}_2$ laser: Nonlinear resonance route and precursors

We investigate the occurrence of extreme and rare events, i.e., giant and rare light pulses, in a periodically modulated ${\mathrm{CO}}_2$ laser model. Due to nonlinear resonant processes, we show a scenario of interaction between chaotic bands of different orders, which may lead to the formation of extreme and rare events. We identify a crisis line in the modulation parameter space, and we show that, when the modulation amplitude increases, remaining in the vicinity of the crisis, some statistical properties of the laser pulses, such as the average and dispersion of amplitudes, do not change much, whereas the amplitude of extreme events grows enormously, giving rise to extreme events with much larger deviations than usually reported, with a significant probability of occurrence, i.e., with a long-tailed non-Gaussian distribution. We identify recurrent regular patterns, i.e., precursors, that anticipate the emergence of extreme and rare events, and we associate these regular patterns with unstable periodic orbits embedded in a chaotic attractor. We show that the precursors may or may not lead to the emergence of extreme events. Thus, we compute the probability of success or failure (false alarm) in the prediction of the extreme events, once a precursor is identified in the deterministic time series. We show that this probability depends on the accuracy with which the precursor is identified in the laser intensity time series.

Figure 1

Phase diagram of the ${\mathrm{CO}}_2$ laser model as a function of modulation parameters. The diagram was computed through the Lyapunov exponents method [51]. Gray shades denote periodic solutions, and yellow-red shades denote chaotic solutions. The dashed line denotes the locus of a crisis between two chaotic attractors (see text for details). Arabic numbers indicate the period of some periodic regions, and roman numbers denote chaotic bands. Modulation parameters $(a,f)$ are $A=(0.045,163.725), B=(0.070,185.650), C=(0.120,200.170)$, and $D=(0.190,208.250)$. The blue arrows in the vertical axis mark $f_{\text{RO}}$ and $2f_{\text{RO}}$, i.e., modulation frequencies associated with the fundamental resonance and the main subharmonic resonance [48, 52], respectively. Frequencies are in kHz.

Figure 2

Amplitude of the laser intensity pulses ($I_A$) as a function of modulation frequency. Arabic numbers denote the period of some periodic orbits, and roman numbers denote some different chaotic bands. The modulation amplitude is fixed in each diagram, and the values are (a) $a=0.005$, (b) $a=0.020$, (c) $a=0.045$, (d) $a=0.070$, and (e) $a=0.190$. Other parameters are not changed. Green arrows denote resonant solutions, i.e., the largest intensity responses for a sequence of different periodic orbits. The blue arrow [in (c)] denotes a crisis without EEs, the red arrow [in (d)] denotes a crisis with EEs and the purple arrow [in (e)] denotes a crisis with SEEs.

Figure 3

Temporal behavior of the laser intensity (a) and the respective PDF (b). PDF of the pulse amplitude (c). Solid lines mark EE definitions, and numbers correspond to how many standard deviations exceed the average of the events. The dashed line marks another EE definition based on $\mathrm{AI}=2$. Modulation parameters correspond to point $A$ in Fig. 1 ($a=0.045,f=163.725\mathrm{kHz}$).

Figure 4

Left column: PDFs of the pulse amplitude. The green lines denote the average amplitude ($\langle I_A\rangle$). Solid lines mark the EE definitions, and numbers correspond to how many standard deviations exceed $\langle I_A\rangle$. Modulation parameters $(a,f)$ are (a) $(0.070,185.65\mathrm{kHz})$, (b) $(0.120,200.17\mathrm{kHz})$, and (c) $(0.190,208.25\mathrm{kHz})$, corresponding to points $B, C$, and $D$ in Fig. 1, respectively. Right column: The respective time series of the laser intensity.

Figure 5

(a) Maximum amplitude of EE, average amplitude ($\langle I_A\rangle$), and its standard deviation (${\sigma }_A$) when the modulation parameters are varied slightly (50 Hz) below the crisis line. (b) Number of standard deviations (${\sigma }_A$) exceeding the average amplitude of the laser pulses when the modulation parameters are varied over the crisis line. The red-dotted line and purple-dotted line mark the $4\sigma$ and $12\sigma$ criterion, respectively. “No EE,” “EE,” and “SEE” stand for “without extreme event,” “extreme event,” and “superextreme event,” respectively, and they illustrate ranges of modulation amplitudes with increasing amplitudes for the observed rare events.

Figure 6

The coefficient of variation ($C_V={\sigma }_A/\langle I_A\rangle$) as a function of modulation frequency, when crossing the critical transition (crisis) for $a=0.19$. The dashed line marks the crisis location ($f\approx 208.27$ kHz).

Figure 7

(a) Temporal behavior of the laser intensity. The box on the left is magnified in (b) and the box on the right is magnified in (d). Parts (c) and (e) show only the amplitudes for the magnified boxes, respectively. The blue and red boxes are discussed in the text. The red lines denote the EE definition, according to the $4\sigma$ criterion. Modulation parameters correspond to point $B$ in Fig. 1 ($a=0.07,f=185.65\mathrm{kHz}$).

Figure 8

Superposition of 200 time series (a) showing a regular pattern preceding an EE and (b) showing the same regular pattern without the emergence of an EE. Only the amplitudes of the laser pulses are plotted. $T$ denotes the index of the successive local maxima of laser intensity. The color scale shows how many times each amplitude bin is visited (see text for details). Modulation parameters are as in Fig. 7.

Figure 9

(a) Time traces of two unstable period-5 orbits of laser intensity (magenta and blue lines) and (b) projection in phase space $(I,N)$. (c) Time trace of an unstable period-3 orbit of laser intensity (red line) and (d) projection in phase space $(I,N)$. In gray, in the panels (b) and (d), is shown the superposition of 50 visitations along POP5 and POP3, respectively. Modulation parameters are as in Fig. 7.

Figure 10

In blue: number of precursors identified leading to the emergence of an EE. In red: number of precursors identified corresponding to false alarms, i.e., not leading to the emergence of an EE. The number of precursors (NP) identified in each case is normalized to the total number of local maxima ($N_{\text{tot}}=7.4\times {10}^8$) of each spanned time series. $\mathrm{\Delta }$ is proportional to the precision in which a precursor is identified (see text for details). The modulation amplitude is fixed at $a=0.07$. The modulation frequency is varied in each panel: (a) $f=185.650\mathrm{kHz}$ (corresponding to point $B$ in Fig. 1), (b) $f=185.660\mathrm{kHz}$, (c) $f=185.665\mathrm{kHz}$, and (d) $f=185.669\mathrm{kHz}$.

Figure 11

Bifurcation diagrams illustrating some differences in amplitudes for successive branches of intensity solutions. (a) Strongly resonant case ($f\approx f_{RO}$), where $f=85$ kHz. (b) Weakly resonant case ($f\approx 2f_{RO}$), where $f=170$ kHz. Roman numbers denote different chaotic bands.

Figure 12

Return maps of the laser pulse amplitude. Black points correspond to the chaotic attractor and red points to unstable period-3 orbit. The lines and numbers are as in Fig. 4. Modulation parameters $(a,f)$ are (a) (0.07, 189.000 kHz), (b) (0.07,185.650 kHz), (c) (0.12,200.170 kHz), and (d) (0.19,208.250 kHz). Each return map was done with 20 000 pulse amplitudes.