Separation of coexisting dynamical regimes in multistate intermittency based on wavelet spectrum energies in an erbium-doped fiber laser

We propose a method for the detection and localization of different types of coexisting oscillatory regimes that alternate with each other leading to multistate intermittency. Our approach is based on consideration of wavelet spectrum energies. The proposed technique is tested in an erbium-doped fiber laser with four coexisting periodic orbits, where external noise induces intermittent switches between the coexisting states. Statistical characteristics of multistate intermittency, such as the mean duration of the phases for every oscillation type, are examined with the help of the developed method. We demonstrate strong advantages of the proposed technique over previously used amplitude methods.

Figure 1

Dynamics of EDFL demonstrating noise-induced multistate intermittency. For noise intensity $\eta =0.23$, two different dynamical regimes (namely, period-1 and period-3) alternate: (a) time series of $x\left(t\right)$, (b) wavelet surface $\left|W\right(s,t\left)\right|$, and (c) dependence of wavelet spectrum energies $w_{1,3}\left(t\right)$ on time $t$. The Roman numerals show the regimes with different oscillation periods. The threshold ${\mathrm{\Delta }}_3$ used for the amplitude criterion is shown by the dot-dashed line in (a). The characteristic time scales for these regimes, $s_{1,3}\pm \mathrm{\Delta }s$, are marked by the dashed lines in (b).

Figure 2

Illustration of the efficiency of the proposed method for noise intensity $\eta =0.47$ when three dynamical regimes, period-1, period-3, and period-4 with frequencies $f_1, f_3$, and $f_4$, alternate with each other. (a) Fragment of analyzed time series $x\left(t\right)$, (b) wavelet surface $W(s,t)$, and (c) dependence of wavelet spectrum energies $w_{1,3,4}\left(t\right)$ on time $t$. The thresholds ${\mathrm{\Delta }}_3$ and ${\mathrm{\Delta }}_4$ used for the amplitude criterion are shown by the dashed lines.

Figure 3

Illustration of the efficiency of the proposed method for noise intensity $\eta =0.97$ when three dynamical regimes, period-3, period-4, and period-5 with frequencies $f_3, f_4$, and $f_5$, alternate with each other. (a) Fragment of analyzed time series $x\left(t\right)$, (b) wavelet surface $W(s,t)$, and (c) dependence of wavelet spectrum energies $w_{3,4,5}\left(t\right)$ on time $t$. The thresholds ${\mathrm{\Delta }}_3, {\mathrm{\Delta }}_4$, and ${\mathrm{\Delta }}_5$ used for the amplitude criterion are shown by the dashed lines.

Figure 4

Probability to detect dynamical regime $P_i$ ($i=1,3,4,5$) vs noise intensity $\eta$.

Figure 5

Probability to detect dynamical regime $P_i$ ($i=1,3,4,5$) as a function of time-series length $L$ for every regime obtained for noise intensity $\eta =0.97$.

Figure 6

Mean residence time $\langle l\rangle$ vs noise intensity $\eta$ for every detected regime.

Figure 7

Probability distributions for lengths of period-1, period-3, period-4, and period-5 dynamical regimes with main frequencies $f_1, f_3, f_4$, and $f_5$, respectively, obtained for noise intensity (a) $\eta =0.23$, (b) $\eta =0.47$, and (c) $\eta =0.97$.