Open Access
Add to BibSonomy
Abstract
In this study, we report a rich variety of large-intensity pulses exhibited by a Zeeman laser model. The instabilities in the system occur via three different dynamical processes, such as quasiperiodic intermittency, Pomeau-Manneville intermittency, and the breakdown of quasiperiodic motion to chaos followed by an interior crisis. This Zeeman laser model is more capable of exploring the major possible types of instabilities when changing a specific system’s parameter in a particular range. We exemplified distinct dynamical transitions of the Zeeman laser model. The statistical measures reveal the appearance of the low probability of large-intensity pulses above the qualifier threshold value. Moreover, they seem to follow an exponential decay that shows a Poisson-like distribution. The impact of noise and time delay effects have been analyzed near the transition point of the system.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The origin of instability in laser models and their different dynamical transitions have been extensively studied [1–5]. However, it is still an active topic of research due to its remarkable properties, and its applications involved in many fields of science and engineering [6]. Notably, in 2007 Solli et al. [7] reported a peculiar type of instability in optical systems: so-called rogue waves or extreme events which are triggered by local instabilities in the phase space of the system. They are characterized by large, yet intermittent, increases of oscillations’ amplitude. The seminal work of Solli et al. [7] encouraged the researchers to explore an unusual type of instability that appears in a different fields of science including optical systems [8–17]. Over the last few decades, attempts have been made in a vast number of deterministic and stochastic mathematical models [18–27] as well as real-time experiments [28–30] to discern their underlying mechanisms and distinct transitions of different types of rare large-amplitude events. Explicitly, the rare and large-amplitude events exhibit different characteristic features compared to the nominal chaos, from the perspective of dynamic, as well as statistical analysis. The ultimate goal of all these studies is to exemplify a generic mechanism of such rare and large-intensity pulses. However, still, it’s a challenging research task that needs further rigorous understanding in this vein to develop an effective tool for their earlier prediction.
Rogue waves or extreme events have been reported in numerous classes of laser models [31–37]. The loss modulated $\mathrm {CO_2}$ laser exhibits successive period-doubling route to chaos followed by extreme and superextreme events owing to the interior-crisis induced intermittency dynamics [33]. In a driven class-B laser, extreme intensity pulses appear via stick-slip dynamics [34]. The transition between low-frequency fluctuation and coherence collapse has been studied in a laser diode, in which extreme events appeared in the low-frequency fluctuation regimes [35]. The optical rogue wave appeared in an injected semiconductor laser, and its formation has been explored through the external crisis-like process [36]. The effect of phase-conjugate and time-delay feedback plays a crucial role in the emergence of a large number of extreme events, as it has been illustrated in a diode laser [37]. Furthermore, the impact of noise and time delay effect in the formation of rare and extreme pulses have been studied in various laser models [36,38–42].
Interestingly, in the majority of the systems which exhibit large-amplitude events is connected with one of the following types of instabilities: interior-crisis-induced intermittency [17,29,43], Pomeau-Manneville (PM) intermittency [29,30,44], quasiperiodic intermittency [45], and breakdown of quasiperiodic motion [16,45]. On the other hand, noise-induced instability between coexisting states triggers large-intensity pulses in an erbium-doped fiber laser [46]. Intermittent switching instability between the order and the turbulent state was found in spatially extended dynamical system [47]. The emergence of intermittent instability-induced rare events with the effect of external stochastic excitation has been reported in the parametrically excited mechanical system [48]. In the coupled neuron models rare large-amplitude bursting appears owning to instability in in-phase and antiphase synchronization [16,49].
The above-mentioned models exhibit only a specific type of instability during their transition either from regular (laminar) or irregular (turbulence) states to rare intermittent large-amplitude events. However, in this study, we exemplified a complex seven-dimensional Zeeman laser [45,50] which manifests all the possible types of instability in the same model, as the pumping parameter of the system is varied in a specific range. Recently, we reported that occurrence of extremely large pulses in the Zeeman laser model originated due to the instabilities in quasiperiodic motion and its intrinsic relation with the appearance of hyperchaotic dynamics [45,51] for the different sets of system parameters. In addition, we have found that the Zeeman laser model is more adequate for exploring the major possible types of instabilities such as quasiperiodic intermittency, PM intermittency, and breakdown of quasiperiodic motion to chaos and then rare large-intensity pulses in a specific range of system parameters. We explained more details of distinct dynamical transitions of the system using bifurcation diagrams and the largest Lyapunov exponents. Besides, the emergence of different large-intensity pulses are characterized by significant height threshold measurement, kurtosis, rare events’ ratio, and distinct probability distribution functions.
The structure of this paper is presented as follows. Section 2 explains the details of the model system, as well as the existence of different instabilities, and the counts of rare events in a wide range of the system parameters. The emergence of occasional large-intensity pulses via quasiperiodic intermittency, PM intermittency, and breakdown of quasiperiodic motion to chaos accompanied by interior-crisis, are illustrated with use of different dynamical as well as statistical measurements in Sections 3, 4 and 5. The impact of noise and time delay effects is illustrated in Section 6. The overall conclusion of our results is given in the final section.
2. Model of Zeeman laser
The dimensionless form of the Zeeman laser model can be expressed by the following differential equations [45,50]
Here, state variables $E_x$ and $E_y$ signify the linear polarization components of the electric field. Similarly, $P_x$, $P_y$ and $D_x$, $D_y$ represent the polarization and atomic inversion, which are related to the transition $| J=1, J_i = 0\rangle \leftrightarrow |J=0\rangle$. The variable $Q$ is proportional to the coherence between upper sub levels $| J=1, J_x = 0\rangle$ and $| J=1, J_y = 0\rangle$. Here, $J$ manifests the atomic transition. System parameter $r$ denotes the incoherent pumping rate, $\sigma$, and $\alpha \sigma$ represent the cavity losses along the $x$ and $y$ directions (see Ref. [45,50] for details). The laser intensity is defined as $I = E_x^2 + E_y^2$. We used the standard $4^{th}$ order Runge-Kutta algorithm with the step size of 0.001 to simulate system dynamics after removing enough transient. For the specific choice of the system parameters $\alpha$, $\sigma$, by varying the pumping rate $r$, we identified that the Zeeman laser exhibits various periodic, quasiperiodic, and chaotic dynamics including large-intensity pulses (LIP) owing to the appearance of distinct types of instabilities in the system.
In order to determine all the possible types of instabilities in the Zeeman laser model, we discriminate distinct states of the system based on their respective largest Lyapunov exponents. The largest Lyapunov exponents of the system were estimated using the perturbation method, more detail on the calculations is reported in Ref. [45,52]. The different types of instabilities with the response of varying the system parameters $r$ and $\alpha$ for the fixed value of $\sigma$ = 6.0 are presented in Fig. 1(a). The different colors in Fig. 1(a) signify periodic (yellow), quasiperiodic (red), and chaotic (gray) states, respectively. Specifically, when we vary pumping parameter $r$ along the horizontal dashed black line at $\alpha$ = 4.0 in Fig. 1(a), the system depicts distinct types of large-intensity events which originate due to the dynamical process of quasiperiodic intermittency, PM intermittency, and torus breakdown to chaos followed by interior-crisis. A closer analysis of the chaotic dynamics in Fig. 1(a) unveils that the system manifests rare LIP in a specific range of parameter regions. To distinguish LIP from the chaos states, we used the significant height threshold ($H_s$) measurement. The $H_s$ value is defined as $H_s$ = $\langle I_n\rangle$ + $6\sigma _I$, where $I_n$ = $I_{max}$ is the peaks of the laser intensity and $\sigma _I$ is its corresponding standard deviation.
Fig. 1. (a) Two parameter phase diagram in $r$ versus $\alpha$ plane show distinct types of instabilities in the Zeeman laser model for the fixed parameter $\sigma$ = 6.0. In (a) yellow, red, and gray regions represent the periodic state (P), quasiperiodic (QP), and chaotic (C) states, respectively. In (b) LIP distinguished from the chaotic states (the gray region in (a)) by calculating the number of large-intensity pulses above a significant height threshold (LIP counts) along with the response of system parameters. The color bar in (b) demonstrates the variation of LIP counts in a range of parameter regions presented on a log scale. To provide more detailed explanation of the different transitions to large-intensity pulses, a specific range of pumping parameters marked by a dashed line at $\alpha$ = 4.0 in (a) is selected.
To measure the number of rare events within a given time length of the laser intensity pulse, we define the quantity ‘count’ as the number of pulses of the laser intensity $I$, whose amplitude exceeds the $H_s$ threshold. The strict, mathematical definition of this quantity requires usage of the Dirac delta function $\delta (x)$ together with the Heaviside step function $\mathbbm{1}(x)$. The integral of $\delta (x)$ attains the value $1$ when $x=0$ and remains $0$ otherwise, such behavior is referred to as “filtering property" of the Dirac delta function [53]. On the other hand, the step function $\mathbbm{1}(x)$ attains the value $1$ when $x > 0$ and is equal $0$ otherwise. In particular, here we assume $\mathbbm{1}(0) = 0$. Using these tools, the ‘count’ quantity can be defined as follows
The first term under the integral equals $1$ if and only if $I$ is increasing. Integral of the other term equals $1$ at each point in which $I = H_s$. Thus, the whole integral is the number of points in the interval $[t_0, t_1]$ in which both: $I = H_s$ and $I$ is increasing. In such manner, the number of pulses, whose amplitude exceeds the $H_s$ threshold, is obtained.
It is clearly shown in Figs. 1(a) and 1(b) that LIP appears from the chaotic states (gray region in Fig. 1(a)) and the counts of LIP vary with the system parameters $\alpha$ and $r$. The detailed dynamical and statistical properties of different LIP as well as their distinct transitions will be explored in more detail in the upcoming sections.
3. Quasiperiodic intermittency via large-intensity pulses
First, we elaborate on the emergence of large-intensity pulses via the quasiperiodic intermittency route which is quite a rare phenomenon from the dynamical system perspective [45,50]. For this interpretation, we set the parameters of model described in Eqn. (1) as $\sigma$ = 6.0, $\alpha$ = 4.0 for fine-tuning of pumping parameter $r$ in the range $r \in$ (29.51, 29.53), through the dashed black line in Fig. 1(a), we identified different dynamics in the Zeeman laser model. To examine the different transitions in the system, we have drawn a bifurcation diagram and its corresponding largest Lyapunov exponents plot against the pumping parameter $r$, which is presented in Fig. 2(a) and 2(b). For the lower values of $r$, i.e., $r$ = 29.46, the system exhibits period-four oscillations and it continues until $r$ = 29.509. The period-4 (P) oscillations turn into quasiperiodic (QP) motion by gradually increasing the pumping parameter (not shown in the bifurcation diagram Fig. 2(b)), and it is clearly shown in the inset of Fig. 2(b). The first largest Lyapunov exponent ($\lambda _1$ - black) is zero, and the second Lyapunov exponent is negative ($\lambda _2$ - red) manifesting that the system is in a periodic state. The system exhibits QP motion, when $\lambda _1$ and $\lambda _2$ become zero at the critical parameter $r$ = 29.51. Further, for the $r$ value larger than 29.5164, the quasiperiodic motion directly transits to rare large-intensity pulses proved as a sudden increase in the amplitude of the peaks laser intensity ($I_{max}$) is portrayed in the bifurcation diagram of Fig. 2(a). Similarly, the largest Lyapunov exponents exemplify positive value in Fig. 2(b) denoted the emergence of complex dynamics of quasiperiodic intermittency with occasional large-intensity pulses (cf. Figure 3(e)). The horizontal solid red line in Fig. 2(a) signifies extreme events qualifier threshold $H_s$. The laser intensities for a range of $r \in$ (29.5165, 29.53) exceed the significant height threshold are rare large-intensity events.
Fig. 2. Quasiperiodic intermittency route: (a) bifurcation diagram for maxima of laser intensity ($I_{max}$) against the pumping parameter $r$ for the fixed system parameters as $\sigma$ = 6.0 and $\alpha$ = 4.0. The red line in (a) manifests the significant threshold $H_s$. LIP appearance directly from the quasiperiodic motion is proved by sudden increase of $I_{max}$ in the bifurcation diagram from the bounded quasiperiodic motion (a), and a discontinuous change in the largest Lyapunov exponent $\lambda _1$ from zero to positive value (b). The emergence LIP directly from the quasiperiodic motion at a critical parameter $r$ = 29.5165 is illustrated in the time evolution of Fig. 3(e). Inset in (b) shows that the system is in periodic state ($\lambda _1$ = 0 and $\lambda _2$ < 0) for $r \in$ (29.46,29.509).
Fig. 3. Time series for different laser intensities (left panels) and its equivalent return map (right panels): (a) and (b) period-4 oscillation, (c) and (d) 4-Torus, (e) and (f) large-intensity pulses originated from quasiperiodic motion for $r$ = 29.509, 29.516, and 29.5165. Red lines in the time series plots (left panels) represent the $H_s$ qualifier threshold.
For the better interpretation of different transitions in the quasiperiodic intermittency route, we displayed time series and return maps for the various values $r$ as shown in Fig. 3(a)-(f). The temporal dynamics of laser intensity for $r$ = 29.509 is presented in Fig. 3(a), which exhibits period-4 oscillation and its return map (cf. Figure 3(b)) signifies four points due to period-4 oscillation. Further, the time evolution of QP for $r$ = 29.516 is depicted in Fig. 3(c), and it is verified by the return map of Fig. 3(d), which exhibits four closed curves. The magnified region of QP is clearly shown in the inset of Fig. 3(d). The time series and return map for $r$ = 29.5165 is illustrated in Fig. 3(e) and 3(f) exemplify the emergence of LIP from the quasiperiodic motion. The temporal dynamics of LIP (cf. Figure 3(e)) proves the appearance of intermittent large-intensity pulses from the laminar phase of quasiperiodic motion and a few of them cross the $H_s$ threshold line (dashed red line). Besides, as it is depicted in Fig. 3(f), the large-intensity pulses (scattered point) occurred from the bounded quasiperiodic motion (see inset).
We have plotted probability distribution function (PDF) for the peaks of laser intensity $\mathrm {I_{max} = I_n}$ for a very long period of time (t-span length as $5.0\times 10^{9}$), which is presented in Fig. 4(a). It shows a heavy-tail distribution along with the extreme pulses over the significant height (solid red vertical line), which slowly decays with the height of the events. The dashed black line in Fig. 4(a) exemplifies a linear fit of exponential decay ($\mathrm {P(I) = ae^{-bI}}$) of PDF over the significant height threshold, which signifies Poisson-like distribution and the fitting parameter values are $a = 0.229$ and $b = 0.3225$.
Fig. 4. (a) Probability distribution function of $I_n$ for the quasiperiodic intermittency at $r$ = 29.5165, signifies a tail of distribution. The vertical solid red line denotes $H_s$ threshold value, the rare events above this threshold manifest an exponential decay which is represented as the dashed black line. (b) Variation of kurtosis values (open red dots) and $\mathrm {EE_{ratio}}$ (closed blue dots) for the pumping parameter range $r \in (29.51, 29.53)$.
Besides, we have used two more indicators other than PDF such as kurtosis and extreme events ratio ($\mathrm {EE_{ratio}}$) to evaluate the rarity of LIP [54] with a response of pumping parameter r, which is presented in Fig. 4(b). The fourth-order moment of the probability density function is called kurtosis and it defined as follows,
4. Large-intensity pulses via PM intermittency route
In addition, when varying the pumping parameter $r$ in the range (31.59, 31.63) along the horizontal dashed line in Fig. 1(a), the Zeeman laser model exhibits LIP which originate via PM intermittency route [57]. To explain the different transitions in the PM intermittency via LIP, one parameter bifurcation diagram is drawn against the pumping parameter $r$ as shown in Fig. 5(a). For a wide range of $r \in$ (31.6153, 31.63), the Zeeman laser exhibits period-15 (P15) oscillations. At the critical parameter value of $r$ = 31.6152, P15 dynamics switches into large-intensity pulses via PM intermittency route. Especially, from the laminar phase of the P15 state, intermittent turbulent phase (chaotic bursting) occurs in an infrequent time interval, and some of the LIP cross the qualifier threshold $H_s$. The transition from periodic to LIP is shown in the largest Lyapunov exponents plot of Fig. 5(b). For the periodic state, $\lambda _1$ is zero and $\lambda _2$ shows a negative value. However, at the transition point, $\lambda _2$ becomes zero and $\lambda _1$ signifies the positive value representing the appearance of complex dynamics in the system. The horizontal red line ($H_s$ threshold) in Fig. 5(a) manifests the visual interpretation of the existence of regular dynamics (P15 blow the $H_s$ threshold) and LIP which exceed the qualifier threshold value.
Fig. 5. PM intermittency via LIP: (a) Bifurcation diagram of laser intensity against the pumping parameter $r$. LIP originated directly from the laminar phase of periodic motion (P15). (b) Plot of Largest Lyapunov exponents ($\lambda _{1,2}$ - solid and dashed lines) shows the transition from periodic state to LIP. The horizontal red line in (a) denotes the $H_s$ threshold of $\langle I_n\rangle$ + $6\sigma _I$.
The time series of P15 is illustrated in Fig. 6(a) for the pumping parameter value $r$ = 31.6153. It shows laminar phases of period fifteen oscillation and its equivalent return map of Fig. 6(b) in the $I_n$ versus $I_{n+1}$ plane proves 15 points, which confirms the appearance of P15 oscillation. Moreover, for the critical parameter value of $r$ = 31.6152, the intermittent turbulent phases occur from the laminar phase (P15 oscillation). Its time series, and return map as shown in Fig. 6(c) and 6(d). In the time series plot of Fig. 6(c), a few LIP exceed the $H_s$ line which is marked in red line. Besides, the return map shows rare bursting appearing from the regular periodic motion and its inset represents the distraction of points from the regular motion (dense points). Further, we analyzed the probability distribution function of LIP which originated from PM intermittency and is depicted in Fig. 7(a). It signifies a heavy-tail distribution. LIP above the significant height threshold manifests an exponential decay ($\mathrm {P(I) = ae^{-bI}}$) which is plotted as the dashed line in Fig. 7(a) and the fitting parameters are $a = 0.6716$ and $b = 0.3064$. Moreover, Fig. 7(b) manifest kurtosis value > 3.0 for the LIP region and $\mathrm {EE_{ratio}}$ is small for parameter region which is closer to the transition point and it is almost saturated for the lower values of $r$ (31.61, 31.59).
Fig. 6. Temporal evolution (left panels) and their equivalent return map (right panels) in the PM intermittency region: (a) and (b) period-15 oscillation for $r$ = 31.6153, (c) and (d) turbulent motion originated from the laminar phase of P15 oscillation for $r$ = 31.6152. Extreme events qualifier threshold ($H_s$) denoted as dashed red lines in the time series (left panels).
Fig. 7. (a) Probability distribution of function LIP for the pumping parameter $r$ = 31.6152 manifest a heavy-tail distribution. Rare events which exceed the significant height threshold (solid red line) exemplify an exponential decay denoted by a dashed black line, and the fitting parameters are given in the text. (b) Kurtosis values (open red dots) and $\mathrm {EE_{ratio}}$ (closed blue dots) change with the response of pumping parameter $r$.
5. Large-intensity pulses via quasiperiodic breakdown to chaos
Next, we explain the emergence of LIP through quasiperiodic breakdown to chaos. For this illustration, once again we focus on the dashed black line of the two-parameter phase diagram of Fig. 1(a) in a specific region pumping parameter $r \in$ (31.74, 33.55). In this region, the Zeeman laser model exhibits different types -> dynamics: of periodic, quasiperiodic, chaos, as well as LIP. In order to explain the distinct dynamical transitions, we have drawn a one-parameter bifurcation diagram by varying $r$ in the range $r \in$ (33.55, 31.89) as shown in Fig. 8(a). In this case for the value of $r$ = 33.415, the system exhibits period-3 oscillations. When we gradually reduce the $r$ value, the period-3 oscillation turns into three torus via Neimark-Sacker bifurcation [58] for $r \approx$ 33.415. As the $r$ value further decrease, 3-torus starts slowly destroying and becomes chaotic motion for $r \approx$ 31.89. Besides, when the pumping parameter turns into the critical value i.e. $r$ = 31.884, the system manifests large-intensity pulses after torus breakdown to chaotic motion. The formation of successive tours-doubling and its breakdown to chaos is clearly shown in Fig. 8(a). The LIP starts appearing for the pumping parameter $r <$ 31.884 and it continues to exist up to $r \approx$ 31.78. The emergence of these different transitions is portrayed in the bifurcation diagram of Fig. 8(b), and it corroborates the largest Lyapunov exponent as shown in Fig. 8(c). In between the pumping, parameter range $r \in$ (31.884, 31.78), the system manifests small periodic windows which we are not focused on in this study. After that, there exists a sudden drop in the laser intensity and the system depicts periodic oscillations. Upon decreasing the pumping parameter $r <$ 31.78, once again the periodic states shifted to higher-order quasiperiodic motion followed by LIP.
Fig. 8. (a) One parameter bifurcation diagram of laser intensity against $r$ shows torus destruction to chaos and (b) LIP appears via after tours-breakdown to chaos as well as quasiperiodic intermittency routes. The red line in (b) represents the significant height threshold ($H_s$). (c) The largest Lyapunov exponents plot confirms the existence of periodic, quasiperiodic, and chaotic states, respectively.
To illustrate distinct dynamics in the range of $r \in$ (31.74, 33.55), we have plotted different temporal evolutions of laser intensity and its equivalent return maps for the various values of $r$ are presented in Fig. 9. The period-3 oscillations for $r$ = 33.415 is shown in Fig. 9(a) and its return map manifests three points (cf. Figure 9(b)). The time series of three torus for $r$ = 33.170 is presented in Fig. 9(c) and its corresponding return map exhibits three closed curves which reveal the appearance of quasiperiodic motion as shown in Fig. 9(d). Further decreasing the $r$ values cause torus breakdown to chaos in the system, and the temporal dynamics of torus breakdown to chaos in the system for $r$ = 31.89 and its appropriate return map are shown in Figs. 9(e) and 9(f). Furthermore, LIP originated after the chaos for $r$ = 31.884 as shown in the temporal dynamics (Fig. 9(g)) and return map (Fig. 9(h)). The time series plot of Fig. 9(e) signifies chaotic states, which do not cross the $H_s$ threshold line. However, when the pumping parameter reaches the critical point, i.e. $r$ = 31.884, the rare extreme pulses appear from the bounded chaotic motion, time series and return map are depicted in Figs. 9(g) and 9(h). Here the existing LIP from chaotic states crosses the extreme events qualifier threshold. Moreover, the return map of Fig. 9(h), clearly shows the appearance of rare events (scatter points) from the chaotic motion (dense region). Further, the system exhibits another periodic attractor in the range $r \in$ (31.78,31.771) which is not shown here. However, for the pumping parameter value $r <$ 31.771, the periodic oscillations turn into higher-order quasiperiodic motion. The time series of higher order quasiperiodic motion and its return map for $r$ = 31.770 are shown in Figs. 9(i) and 9(j). The blown-up region in the return map (Fig. 9(j)) depicts closed curves confirming the existence of quasiperiodic motion. Finally, another time series and return map of LIP for $r$ = 31.765 are presented in Figs. 9(k) and 9(l), here the appearance of extreme pulses from the chaotic motion cross the $H_s$ threshold line.
Fig. 9. Temporal dynamics of different laser intensities (left panels) and its equivalent return maps (right panels). (a) and (b) period-3 oscillation, (c) and (d) 3-Torus, (e) and (f) chaos, (g) and (h) large-intensity pulses originated from the bounded chaotic motion, (i) and (j) complex quasiperiodic motion, (k) and (l) rare large-intensity pulses originated from the quasiperiodic motion for $r$ = 33.415, 33.170, 31.89, 31.884, 31.770, and 31.765, respectively. Red lines in the time series plots represent the qualifier threshold value ($H_s$).
We have plotted the PDF for two different LIP for the pumping parameter values $r =$ 31.884, and 31.765 which are depicted in Fig. 10(a) and Fig. 10(b). The PDFs of Fig. 10 manifest heavy-tail distribution and the rare events which exceed the significant height that follows an exponential decay ($\mathrm {P(I) = ae^{-bI}}$). The fitting parameter values for Fig. 10(a) are $a = 0.01054$ and $b = 0.2845$, for Fig. 10(b) are $a = 1.19$ and $b = 0.2858$. Once again, we have calculated kurtosis and $\mathrm {EE_{ratio}}$ for the pumping parameter range $r \in (31.74, 31.85)$ as shown in Fig. 10(c). The system shows the kurtosis value > 3.0 for LIP and less than 3.0 for the other dynamics. On the other hand, the $\mathrm {EE_{ratio}}$ proves LIP has a small ratio near the transition point as compared with other events in a wide range of pumping parameter values.
Fig. 10. Probability distribution functions: (a) for $r = 31.884$ and (b) for $r = 31.765$ signify a heavy-tail distribution and LIP which appears over the $H_s$ threshold (solid red line) proves an exponential distribution. (b) Kurtosis (open red dots) and $\mathrm {EE_{ratio}}$ (closed blue dots) values for the wide range of pumping parameter $r$.
We have explored three different routes to large-intensity pules for the fixed value of $\sigma = 6.0$ and varying the other two system parameters $\alpha$ and the pumping parameter $r$. The obtained results are not restricted to a specific $\sigma$ value, and the system exhibits a distinct transition to large-intensity pulses for the other value of $\sigma$, the obtained results are presented in Appendix.
6. Noise and time delay effect
In this section, we address an obvious question, what are the noise and time delay effects in the Zeeman laser model? The problem is crucial, since these two factors play an important role in the emergence of distinct rare large-intensity pulses. To explore the noise effect in the system, we have included the noise term in the first equation as follows,
Here $\xi$ is Gaussian-white noise with zero mean and variance of one, and $D$ denotes the noise strength. Equation (4) was solved using the Runge-Kutta method based on the Ito algorithm [59].
We illustrate the noise effect in the quasiperiodic intermittency via the LIP region. For that, we have plotted the bifurcation diagram in Fig. 11(a), for two different values of noise strength such as $D = 1.0\times 10^{-4}$ (yellow dots), $D = 1.0\times 10^{-3}$ (gray circles), and varying pumping parameter $r \in$ (29.505, 29.52). The appearance of LIP is robust under the influence of weak noise (yellow dots in Fig. 11(a)). Nevertheless, we identified a tiny shift in the transition point in the system as compared with Fig. 2(a). However, for higher noise strength, the Zeeman laser model exhibits irregular swapping between LIP and quasiperiodic motion near the transition point, as it is clearly shown in Fig. 11(a) (gray circles). The random switching between two different attractors with the influence of noise is known as attractor hopping dynamics which reported in details [46,60,61]. Note that the Zeeman laser model does not exemplified multistability for the consider parameter region. However, we studied the influence of noise by varying the pumping parameter of the system. Furthermore, we have exemplified the probability distributions at the transition point of two different noise intensity levels. We have obtained the heavy-tail distribution as shown in Fig. 11(b). It is inferred from the PDF plot that for lower noise intensity the appearance of LIP is very rare. On the other hand, for a higher noise strength, the probability of the emergence of LIP is increasing as presented in Fig. 11(b). Hence, noise plays a significant role in the formation of large-intensity pulses in the Zeeman laser model.
Fig. 11. (a) Bifurcation diagram for a specific range of pumping parameters for different values of noise intensity $D = 1.0\times 10^{-4}$ (yellow dots), and $D = 1.0\times 10^{-3}$ (gray circles), respectively. (b) PDFs at the transition point for different noise intensities (yellow - $D = 1.0\times 10^{-4},~r = 29.5161$ and gray - $D = 1.0\times 10^{-3},~r = 29.5103$) manifest heavy-tail distributions.
Finally, we demonstrate the impact of time delay in the Zeeman laser model by incorporating a self-feedback delay in the $E_x$ variable of the system. The modified equation governing the time delay is presented below,
The time delay, represented by the symbol $\tau$, and the strength of the delay feedback, represented by the symbol $\epsilon$, play a crucial role in the behavior of the Zeeman laser model. In this study, the strength of the feedback was fixed at a value of 0.01, while the time delay was set to 1.0. The effect of this combination was observed by varying the pumping parameter values in the range of 29.49 to 29.80.
The results, presented in Fig. 12(a), show that the time delay feedback has a noticeable impact on the transition point of large-intensity pulses in the Zeeman laser model. The transition point shifted from 29.5165 to 29.53, demonstrating that time delay feedback can significantly affect the dynamics of the system. However, a more in-depth analysis of the effect of time delay feedback is necessary to fully understand its impact on the Zeeman laser model. This is a topic that deserves further study in the future. Notably, we have included the noise and time delay terms in the first state variable $E_x$ of the Zeeman laser model and explored its distinct dynamical transitions. Furthermore, when examining the effects of noise and time delay in the second state variable $E_y$ or both $E_x$ and $E_y$, the system exhibits similar dynamics. This is because, at a critical pumping parameter value, laser intensity $I$ and the separate state variables $E_x^2$ and $E_y^2$ manifest large-intensity pulses [45].
Fig. 12. (a) Bifurcation diagram for a range of $r$ values for the fixed feedback strength $\epsilon = 0.01$ and delay $\tau = 1.0$ manifest a significant shift in the emergence of LIP. (b) PDF of LIP at the transition ($r$ = 29.53) signifies a heavy-tail distribution.
Finally Fig. 12(b) shows the probability distribution at the transition point ($r$ = 29.53), which exhibits a heavy-tail distribution. This confirms that rare events are less likely to occur, further emphasizing the significance of the effect of time delay feedback on the Zeeman laser model.
7. Conclusion
We have investigated the formation of large-intensity pulses in a Zeeman laser model through different routes such as quasiperiodic intermittency, PM intermittency, and quasiperiodic breakdown to chaos, followed by interior-crisis dynamics, by selecting appropriate system parameters. The distinct dynamical transitions in the system were analyzed using bifurcation diagrams and confirmed by the largest Lyapunov exponents. The presence of large-intensity pulses (LIPs) was verified by significant height threshold measurements. The probability density functions of different LIPs showed a heavy-tail distribution, and the rare events over the significant height threshold demonstrated exponential decay in all cases. Furthermore, we calculated the kurtosis and $\mathrm {EE_{ratio}}$ for different pumping parameter values, which showed that LIPs exhibit kurtosis values greater than three and $\mathrm {EE_{ratio}}$ values that are very rare in the parameter region close to the transition points. Lastly, we demonstrated the impact of noise and time delay feedback on the formation of large-intensity pulses. This Zeeman laser model is suitable for uncovering the different types of instabilities that trigger the system dynamics, either from regular (periodic or quasiperiodic) or chaotic motion, to large-intensity pulses.
8. Appendix
The results of our study show that the transition to large-intensity pulses in the system does not depend on a particular selection of $\sigma$ = 6.0. On the contrary, the Zeeman laser model exhibits similar transitions (all three routes to LIP) for different values of $\sigma$. In this appendix, we provide examples of different formations of large-intensity pulses by fixing the values of $\alpha$ = 4.0 and $\sigma$ = 8.0, and varying the pumping parameter within a specific range.
To explore the quasiperiodic intermittency route to large-intensity pulses (LIP), we vary the pumping parameter $r$ within the range of $r \in$ (29.25, 29.975). We illustrate the transition from periodic to quasiperiodic dynamics using the bifurcation diagram and Lyapunov exponent for the lower range of $r \in$ (29.25, 29.85), as shown in Figs. 13(a) and 13(b). Furthermore, we identified the transition from quasiperiodic dynamics to large-intensity pulses for $r >$ 29.922, which is confirmed by the sudden increase in $I_{\text {max}}$ in the bifurcation diagram shown in Fig. 13(c). Additionally, the transition of the largest Lyapunov exponent from zero to a positive value is portrayed in Fig. 13(d). The red line in Fig. 13(c) represents the significant height threshold $H_s$. Figure 14(a) presents the temporal dynamics of quasiperiodic intermittency for $r$ = 29.923, where rare large-intensity pulses cross the $H_s$ threshold. The return map for quasiperiodic intermittency dynamics, depicted in Fig. 14(b), illustrates the emergence of rare LIP from the bounded region of quasiperiodic dynamics (cf. inset in Fig. 14(b)). Furthermore, the probability density function (PDF) showcases the infrequent occurrence of LIP over a significant height threshold, displaying a heavy-tailed distribution.
Fig. 13. Quasiperiodic intermittency for fixed $\alpha$ = 4.0 and $\sigma = 8.0$: One parameter bifurcation diagram (a) and its equivalent largest Lyapunov exponents (b) exhibit transition from P to QP dynamics in the Zeeman laser model. The abrupt transition from the QP to LIP proved by instant changes in the peak intensity ($I_{max}$) and the transition of largest Lyapunov exponent from zero to positive value. The solid red line in (c) denoted the significant height threshold value.
Fig. 14. (a) Time evolution of quasiperiodic intermittency for $r$ = 29.923 and (b) its equivalent return map confirm the apprentice of rare events from the bounded quasi-periodic motion. Inset in (b) signifies the magnified region of bounded quasiperiodic motion. (c) Probability distribution function of quasiperiodic intermittency manifest a heavy-tail distribution. The horizontal dashed red line in (a) and the vertical solid red line in (c) denoted the significant height threshold.
The transition from quasiperiodic breakdown to chaos, followed by the formation of large-intensity pulses, is illustrated using the bifurcation diagram and largest Lyapunov exponent, as depicted in Figs. 15(a) and 15(b). For $r$ = 34.4, the system exhibits periodic dynamics. Gradually decreasing the pumping parameter $r$, the system undergoes a switch to four tori at $r$ = 34.05. Upon further decreasing $r$, we identified torus breakdown to chaos at $r$ = 33.46, confirmed by the positive transition of the largest Lyapunov exponent from zero in Fig. 15(b). The emergence of LIP is characterized by the abrupt increase in the peak intensity $I_{\text {max}}$ in the bifurcation diagram of Fig. 15(a) at $r$ = 33.25. This is also reflected as a sudden change in the largest Lyapunov exponent in Fig. 15(b). The significant height threshold is represented by the red line in the bifurcation diagram of Fig. 15(a). In the time series of Fig. 15(c) for $r$ = 33.25, we present the appearance of rare events originating from the bounded chaotic motion, where a few LIP exceed the significant height threshold (indicated by the dashed red line). The probability density function in Fig. 15(d) exemplifies LIP originating from chaotic dynamics, exhibiting an exponential decay of the distribution with respect to the peaks of laser intensity.
Fig. 15. Quasiperiodic breakdown to chaos via large expansion for fixed $\alpha$ = 4.0 and $\sigma = 8.0$: One parameter bifurcation diagram (a) and its equivalent largest Lyapunov exponents (b) signifies different transition in the system. (c) Time evolution of LIP which is originated from the bounded chaotic dynamics. (d) PDF shows a heavy-tail distribution. The red lines in (a), (c), and (d) denotes the significant height threshold ($H_s$).
Finally, we illustrate the PM intermittency via large expansion using the bifurcation diagram and Lyapunov exponent presented in Figs. 16(a) and 16(b), respectively. In the range of $r \in$ (38.75, 38.797), the Zeeman laser model exhibits period-three dynamics. However, for the pumping parameter $r \ge$ 38.798, the system undergoes a transition from a bounded laminar state to occasional turbulent bursting, resulting in the emergence of LIP. We observe a sudden large expansion in $I_{\text {max}}$ (see Fig. 16(a)) and discontinuous changes in the first largest Lyapunov exponent (see Fig. 16(b)) during this transition. The inset in Fig. 16(b) indicates that the system is in a periodic state before the large expansion, characterized by largest Lyapunov exponent which are $\lambda _1$ = 0 and $\lambda _2$ less than zero. Notably, both the quasiperiodic-to-chaos transition and the PM intermittency via large expansion exhibit positive values for the two largest Lyapunov exponents, indicating the presence of hyperchaotic dynamics in this specific parameter region. The intrinsic relationship between rare large-amplitude events and hyperchaotic dynamics is discussed in Ref. [26,51]. Figure 16(c) demonstrates the temporal dynamics of PM intermittency for a specific value of $r$ = 38.798, where the rare events surpass the threshold $H_s$ (dashed red line). Furthermore, the probability density function (PDF) in Fig. 16(d) exhibits a heavy-tailed distribution.
Fig. 16. PM intermittency via large expansion for fixed $\alpha$ = 4.0 and $\sigma = 8.0$: One parameter bifurcation diagram (a) and its corroborate largest Lyapunov exponents (b) signifies transition from period three to LIP in the Zeeman laser. Inset in (b) shows zoom in region of first two largest Lyapunov exponent ($\lambda _1$ = 0 and $\lambda _2$ less than zero) signifies the existence the periodic dynamics in the system. (c) Time evolution of LIP which is originated from the laminar phase of period three oscillations for $r$ = 38.798. (d) PDF shows a heavy-tail distribution. The red lines in (a), (c), and (d) denotes the significant height threshold ($H_s$).
Funding
Infrastruktura PL-Grid; Chennai Institute of Technology (CIT/CNS/2023/RP-016); Narodowe Centrum Nauki, Poland, OPUS Program (2017/27/B/ST8/01619, 2018/29/B/ST8/00457, 2021/43/B/ST8/00641).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. J. Ohtsubo, Semiconductor lasers: stability, instability and chaos, vol. 111 (Springer, 2012).
2. J. R. Tredicce, F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B 2(1), 173–183 (1985). [CrossRef]
3. F. Arecchi, “Instabilities and chaos in optics,” Phys. Scr. T23, 160–164 (1988). [CrossRef]
4. L. M. Narducci and N. B. Abraham, Laser physics and laser instabilities (World Scientific, 1988).
5. P. Mandel, “Theoretical problems in cavity nonlinear optics,” Theoretical Problems in Cavity Nonlinear Optics (2005).
6. A. Uchida, Optical communication with chaotic lasers: applications of nonlinear dynamics and synchronization (John Wiley & Sons, 2012).
7. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007). [CrossRef]
8. M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528(2), 47–89 (2013). [CrossRef]
9. Y. Huang, S. Gu, Y. Zeng, P. Zhou, and N. Li, “Spatiotemporal chaos induces extreme events in a three-element laterally coupled laser array,” Opt. Lett. 47(18), 4632–4635 (2022). [CrossRef]
10. K. Hammani, C. Finot, J. M. Dudley, and G. Millot, “Optical rogue-wave-like extreme value fluctuations in fiber Raman amplifiers,” Opt. Express 16(21), 16467–16474 (2008). [CrossRef]
11. A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17(19), 17010–17015 (2009). [CrossRef]
12. C. Metayer, A. Serres, E. Rosero, W. Barbosa, F. De Aguiar, J. R. Leite, and J. Tredicce, “Extreme events in chaotic lasers with modulated parameter,” Opt. Express 22(17), 19850–19859 (2014). [CrossRef]
13. J. Lauterio-Cruz, H. Ibarra-Villalon, O. Pottiez, Y. Bracamontes-Rodriguez, O. Torres-Mu noz, J. Hernandez-Garcia, and H. Rostro-Gonzalez, “Numerical study of complex dynamics and extreme events within noise-like pulses from an erbium figure-eight laser,” Opt. Express 27(26), 37196–37213 (2019). [CrossRef]
14. P. Mu, Y. Huang, P. Zhou, et al., “Extreme events in two laterally-coupled semiconductor lasers,” Opt. Express 30(16), 29435–29448 (2022). [CrossRef]
15. J. Xu, J. Wu, J. Ye, J. Song, B. Yao, H. Zhang, J. Leng, W. Zhang, P. Zhou, and Y. Rao, “Optical rogue wave in random fiber laser,” Photonics Res. 8(1), 1–7 (2020). [CrossRef]
16. A. Mishra, S. L. Kingston, C. Hens, T. Kapitaniak, U. Feudel, and S. K. Dana, “Routes to extreme events in dynamical systems: Dynamical and statistical characteristics,” Chaos 30(6), 063114 (2020). [CrossRef]
17. S. N. Chowdhury, A. Ray, S. K. Dana, and D. Ghosh, “Extreme events in dynamical systems and random walkers: A review,” Phys. Rep. 966, 1–52 (2022). [CrossRef]
18. M. Farazmand and T. P. Sapsis, “A variational approach to probing extreme events in turbulent dynamical systems,” Sci. Adv. 3(9), e1701533 (2017). [CrossRef]
19. M. Farazmand and T. P. Sapsis, “Extreme events: Mechanisms and prediction,” Appl. Mech. Rev. 71(5), 1 (2019). [CrossRef]
20. S. L. Kingston, K. Suresh, K. Thamilmaran, and T. Kapitaniak, “Extreme and critical transition events in the memristor based Liénard system,” Eur. Phys. J. Spec. Top. 229(6-7), 1033–1044 (2020). [CrossRef]
21. A. Ouannas, N. Debbouche, V.-T. Pham, S. L. Kingston, and T. Kapitaniak, “Chaos in fractional system with extreme events,” Eur. Phys. J. Spec. Top. 230(7-8), 2021–2033 (2021). [CrossRef]
22. A. Ray, A. Mishra, D. Ghosh, T. Kapitaniak, S. K. Dana, and C. Hens, “Extreme events in a network of heterogeneous Josephson junctions,” Phys. Rev. E 101(3), 032209 (2020). [CrossRef]
23. S. L. Kingston and T. Kapitaniak, “Intermittent large amplitude bursting in Hindmarsh-Rose neuron model,” in 2021 5th Scientific School Dynamics of Complex Networks and their Applications (DCNA), (IEEE, 2021).
24. S. Kumarasamy, S. Srinivasan, P. B. Gogoi, and A. Prasad, “Emergence of extreme events in coupled systems with time-dependent interactions,” Commun. Nonlinear Sci. Numer. Simul. 107, 106170 (2022). [CrossRef]
25. D. Premraj, K. Suresh, S. A. Pawar, L. Kabiraj, A. Prasad, and R. Sujith, “Dragon-king extreme events as precursors for catastrophic transition,” Europhys. Lett. 134(3), 34006 (2021). [CrossRef]
26. S. L. Kingston, T. Kapitaniak, and S. K. Dana, “Transition to hyperchaos: Sudden expansion of attractor and intermittent large-amplitude events in dynamical systems,” Chaos 32(8), 081106 (2022). [CrossRef]
27. B. Kaviya, R. Gopal, R. Suresh, and V. Chandrasekar, “Route to extreme events in a parametrically driven position-dependent nonlinear oscillator,” Eur. Phys. J. Plus 138(1), 36 (2023). [CrossRef]
28. K. Lehnertz, “Epilepsy: Extreme events in the human brain,” in Extreme events in nature and society, (Springer, 2006), pp. 123–143.
29. S. L. Kingston, K. Thamilmaran, P. Pal, U. Feudel, and S. K. Dana, “Extreme events in the forced Liénard system,” Phys. Rev. E 96(5), 052204 (2017). [CrossRef]
30. A. Mishra, S. Saha, M. Vigneshwaran, P. Pal, T. Kapitaniak, and S. K. Dana, “Dragon-king-like extreme events in coupled bursting neurons,” Phys. Rev. E 97(6), 062311 (2018). [CrossRef]
31. O. Spitz, J. Wu, A. Herdt, G. Maisons, M. Carras, W. E. Elsäßer, C.-W. Wong, and F. Grillot, “Extreme events in quantum cascade lasers,” Adv. Photonics 2(06), 066001 (2020). [CrossRef]
32. S. Coulibaly, M. Clerc, F. Selmi, and S. Barbay, “Extreme events following bifurcation to spatiotemporal chaos in a spatially extended microcavity laser,” Phys. Rev. A 95(2), 023816 (2017). [CrossRef]
33. C. Bonatto and A. Endler, “Extreme and superextreme events in a loss-modulated CO 2 laser: Nonlinear resonance route and precursors,” Phys. Rev. E 96(1), 012216 (2017). [CrossRef]
34. S. Kumarasamy and A. N. Pisarchik, “Extreme events in systems with discontinuous boundaries,” Phys. Rev. E 98(3), 032203 (2018). [CrossRef]
35. D. Choi, M. J. Wishon, J. Barnoud, C. Chang, Y. Bouazizi, A. Locquet, and D. Citrin, “Low-frequency fluctuations in an external-cavity laser leading to extreme events,” Phys. Rev. E 93(4), 042216 (2016). [CrossRef]
36. J. Zamora-Munt, B. Garbin, S. Barland, M. Giudici, J. R. R. Leite, C. Masoller, and J. R. Tredicce, “Rogue waves in optically injected lasers: Origin, predictability, and suppression,” Phys. Rev. A 87(3), 035802 (2013). [CrossRef]
37. É. Mercier, A. Even, E. Mirisola, D. Wolfersberger, and M. Sciamanna, “Numerical study of extreme events in a laser diode with phase-conjugate optical feedback,” Phys. Rev. E 91(4), 042914 (2015). [CrossRef]
38. D. Boukhaoui, D. Mallek, A. Kellou, H. Leblond, F. Sanchez, T. Godin, and A. Hideur, “Influence of higher-order stimulated brillouin scattering on the occurrence of extreme events in self-pulsing fiber lasers,” Phys. Rev. A 100(1), 013809 (2019). [CrossRef]
39. J. A. Reinoso, J. Zamora-Munt, and C. Masoller, “Extreme intensity pulses in a semiconductor laser with a short external cavity,” Phys. Rev. E 87(6), 062913 (2013). [CrossRef]
40. A. K. Dal Bosco, D. Wolfersberger, and M. Sciamanna, “Extreme events in time-delayed nonlinear optics,” Opt. Lett. 38(5), 703–705 (2013). [CrossRef]
41. C.-H. Uy, D. Rontani, and M. Sciamanna, “Vectorial extreme events in VCSEL polarization dynamics,” Opt. Lett. 42(11), 2177–2180 (2017). [CrossRef]
42. K. Panajotov, M. Tlidi, Y. Song, and H. Zhang, “Control of dissipative rogue waves in nonlinear cavity optics: Optical injection and time-delayed feedback,” Chaos 30(5), 053103 (2020). [CrossRef]
43. A. Ray, S. Rakshit, D. Ghosh, and S. K. Dana, “Intermittent large deviation of chaotic trajectory in Ikeda map: Signature of extreme events,” Chaos 29(4), 043131 (2019). [CrossRef]
44. V. Varshney, S. Kumarasamy, A. Mishra, B. Biswal, and A. Prasad, “Traveling of extreme events in network of counter-rotating nonlinear oscillators,” Chaos 31(9), 093136 (2021). [CrossRef]
45. S. L. Kingston, A. Mishra, M. Balcerzak, T. Kapitaniak, and S. K. Dana, “Instabilities in quasiperiodic motion lead to intermittent large-intensity events in Zeeman laser,” Phys. Rev. E 104(3), 034215 (2021). [CrossRef]
46. A. N. Pisarchik, R. Jaimes-Reátegui, R. Sevilla-Escoboza, G. Huerta-Cuellar, and M. Taki, “Rogue waves in a multistable system,” Phys. Rev. Lett. 107(27), 274101 (2011). [CrossRef]
47. P. Galuzio, R. Viana, and S. Lopes, “Control of extreme events in the bubbling onset of wave turbulence,” Phys. Rev. E 89(4), 040901 (2014). [CrossRef]
48. M. A. Mohamad and T. P. Sapsis, “Probabilistic description of extreme events in intermittently unstable dynamical systems excited by correlated stochastic processes,” SIAM/ASA J. on Uncertain. Quantification 3(1), 709–736 (2015). [CrossRef]
49. A. Saha and U. Feudel, “Characteristics of in-out intermittency in delay-coupled Fitzhugh–Nagumo oscillators,” Eur. Phys. J. Spec. Top. 227(10-11), 1205–1219 (2018). [CrossRef]
50. J. Redondo, G. J. de Valcárcel, and E. Roldán, “Intermittent and quasiperiodic behavior in a Zeeman laser model with large cavity anisotropy,” Phys. Rev. E 56(6), 6589–6600 (1997). [CrossRef]
51. S. L. Kingston, M. Balcerzak, S. K. Dana, and T. Kapitaniak, “Transition to hyperchaos and rare large-intensity pulses in Zeeman laser,” Chaos 33(2), 023128 (2023). [CrossRef]
52. M. Balcerzak, D. Pikunov, and A. Dabrowski, “The fastest, simplified method of Lyapunov exponents spectrum estimation for continuous-time dynamical systems,” Nonlinear Dyn. 94(4), 3053–3065 (2018). [CrossRef]
53. D. A. McQuarrie, Mathematical methods for scientists and engineers (University science books, 2003).
54. K. Talouneh, C. Rimoldi, R. Kheradmand, G. Tissoni, and M. Eslami, “Control of spatiotemporal rogue waves by harmonic pump modulation in a semiconductor laser with a saturable absorber,” Phys. Rev. A 102(3), 033508 (2020). [CrossRef]
55. M. Onorato, A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini, and C. Stansberg, “Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments,” Phys. Rev. E 70(6), 067302 (2004). [CrossRef]
56. M. Cristelli, A. Zaccaria, and L. Pietronero, “Universal relation between skewness and kurtosis in complex dynamics,” Phys. Rev. E 85(6), 066108 (2012). [CrossRef]
57. Y. Pomeau and P. Manneville, “Intermittent transition to turbulence in dissipative dynamical systems,” Commun. Math. Phys. 74(2), 189–197 (1980). [CrossRef]
58. Y. A. Kuznetsov, Elements of applied bifurcation theory, vol. 112 (Springer Science & Business Media, 2013).
59. N. J. Newton, “Asymptotically efficient Runge-Kutta methods for a class of Ito and Stratonovich equations,” SIAM J. Appl. Math. 51(2), 542–567 (1991). [CrossRef]
60. G. Huerta-Cuellar, A. N. Pisarchik, and Y. O. Barmenkov, “Experimental characterization of hopping dynamics in a multistable fiber laser,” Phys. Rev. E 78(3), 035202 (2008). [CrossRef]
61. A. N. Pisarchik and A. E. Hramov, Multistability in Physical and Living Systems (Springer, 2022).
