Dragon-kings death in nonlinear wave interactions

TL;DR

This paper demonstrates the existence of dragon-king extreme events in nonlinear three-wave interactions and proposes a method to prevent them by introducing a small perturbing wave.

Contribution

It reveals the presence of dragon-king events in nonlinear wave systems and shows how small perturbations can prevent these catastrophic events.

Findings

Dragon-king events are linked to phase transitions and tipping points.

Adding a small wave can prevent extreme events.

The study enhances understanding of controlling nonlinear wave phenomena.

Abstract

Extreme events are characterised by low probabilities and high impact on the systems. As a consequence, it is possible to find many studies about the predictability and suppression of extreme events. In this work, we show the existence of dragon-kings extreme events in nonlinear three-wave interactions. Dragon-king extreme events, identified by phase transitions, tipping points, and catastrophes, affects fluctuating systems. We show that these events can be avoided by adding a perturbing small amplitude wave to the system.

Equations

$${\dot{A}}_1$$

$${\dot{A}}_2$$

$${\dot{A}}_3$$

$${\dot{A}}_4$$

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footnote **Dragon-kings death in nonlinear wave interactions

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Moises S. Santos1, José D. Szezech Jr1,2, Antonio M. Batista1,2, Kelly C. Iarosz3, Iberê L. Caldas3, Ricardo L. Viana4

1Graduate in Science Program - Physics, State University of Ponta Grossa, 84030-900, Ponta Grossa, PR, Brazil.

2Department of Mathematics and Statistics, State University of Ponta Grossa, 84030-900, Ponta Grossa, PR, Brazil.

3Institute of Physics, University of São Paulo, 05508-900, São Paulo, SP, Brazil.

4Department of Physics, Federal University of Paraná, 80060-000, Curitiba, PR, Brazil.

Corresponding author: moises.wz@gmail.com

Abstract

Extreme events are characterised by low probabilities and high impact on the systems. As a consequence, it is possible to find many studies about the predictability and suppression of extreme events. In this work, we show the existence of dragon-kings extreme events in nonlinear three-wave interactions. Dragon-king extreme events, identified by phase transitions, tipping points and catastrophes, affects fluctuating systems. We show that these events can be avoided by adding a perturbing small amplitude wave to the system.

1 Introduction

Extreme events are by definition scarce and they can have a relevant impact on the systems. These events have been observed in weather [1], optics [2], and plasma physics [3]. In optical rogue waves, extreme events emerge from a turbulent state [4]. Riccardo et al. [5] reported that extreme event is compatible with general halo current trends in magnetically confined plasma physics in the tokamak Joint European Torus (JET).

Extreme events have been estimated through the extrapolation of power law frequency-size distributions [6]. Many extreme events, known as dragon-kings, do not belong to a power law distribution [7]. The dragon-kings were reported by Johansen and Sornette [8] in the distribution of financial drawdowns. Sornette and Ouillon [9] discussed the mechanisms, statistical tests, and empirical evidences of dragon-kings. Dragon-king extreme events was found in neuronal networks by Mishra et al. [10]. They presented evidence of these events in coupled bursting neurons.

Recently, many articles about the suppression of extreme events have been published. Direct corrective reset to the system was used as a suppression method [11]. Cavalcante et al. [12] studied coupled chaotic oscillators and showed that extreme events can be suppressed by means of tiny perturbations. Small perturbations were used to reduce the appearance of extreme events in damped one-dimensional nonlinear Schrödinger equation [13]. Krüger et al. [14] demonstrated that the existence of extreme events can be controlled by means of the properties of the phase space in Hamiltonian systems.

We analyse a nonlinear three-wave interactions model. The nonlinear three-wave interaction is the lowest order effect in systems described by waves superposition [15]. It plays an important role in nonlinear optics, plasma physics and hydrodynamics [16]. Chian and Abalde [17] developed a nonlinear theory of three-wave interactions of Langmuir waves with whistler waves in the solar wind. Batista et al. [18] considered three-wave interaction to investigate drift-wave turbulence in tokamak edge plasma. In our simulations, we identify Lorentzian pulses. These pulses have been observed in magnetised plasmas and interactions of drift-Alfvén waves [19]. They can appear in flow trajectories due to topological alterations and in chaotic orbits [20, 21].

We calculate the frequency-size distributions of the wave amplitude in the three-wave interactions model. Depending on the parameter, the distribution follows a power law except for large sizes. This behaviour indicates the existence of dragon-kings extreme events. Aiming to suppress the dragon-kings, we propose a fourth wave as a control method. Batista et al. [22] used a fourth resonant wave of small amplitude to control chaotic behaviour of three interacting modes. Coexistence of attractors was observed in a dissipative nonlinear parametric four-wave interactions [23]. In this work, we show that a fourth wave is able to kill the dragon-kings in nonlinear three-wave interactions.

The paper is organised as follows: In Section $2$, we describe the nonlinear three-wave interactions model. Section $3$ shows the dragon-kings extreme events and presents our method to suppress these events. In the last Section, we draw our conclusions.

2 Nonlinear tree-wave interactions model

The nonlinear three-wave interactions are described by three first-order differential equations [24, 25, 26]. We include a fourth wave interacting with the first and second waves [23]. The equations of the four-wave interactions are given by

[TABLE]

where $A_{i}$ is the wave amplitude ($i=1,2,3,4$), $A^{*}_{i}$ denotes the conjugate complex, $\nu_{1}$ is the energy injection coefficient, $\nu_{i}$ ($i=2,3,4$) is the dissipation parameter, and $\delta_{3,4}$ are a small mismatch. We consider $\nu_{1}=1$, $\nu_{2}=\nu_{3}=\nu$, $\delta_{4}=0$, $\nu_{4}=-0.83$, transient time equal to $10^{4}$, and random initial condition in the interval $\left[0,1\right]$ for $A_{i}$, and $A^{*}_{i}$. The $r$ parameter controls the intensity of the fourth wave, at the limit when $r=0$, the four-wave interaction model becomes the three-wave interaction model.

The temporal evolution of waves $|A_{1}|$ and $|A_{2}|$ for the three-wave model is exhibited in Fig. 1(a) and Fig. 1(b), respectively. The time evolution of wave $|A_{3}|$ is similar to $|A_{2}|$, so it was not shown in Fig. 1. We consider $\delta_{3}=2.2$, $\nu=-14.5$, and $r=0$. As a result, the system exhibits chaotic behaviour, namely the dynamical system is sensitive to initial conditions. The chaotic attractor is plotted in Fig. 2(a). The attractor has a large number of periodic orbits identified by varying the parameter $\delta_{3}$. Figure 2(b) shows a period-4 stable periodic orbit for $\delta_{3}=1.1$ and $\nu=-14.5$.

The short time interval of the temporal evolution (pulse) of $A_{2}$ (Fig. 1(b)) can be fitted by a Lorentzian function, as shown in Fig. 3(a) (red line) for the same values of Fig. 1. Lorentzian pulses have been associated with chaotic behaviour [20, 21]. In fact, the chaotic behaviour for theses parameters is seen in Fig. 3(b) where is plotted the bifurcation diagram for the maximum value of $|A_{2}|$ as a function of $\delta_{3}$. The system displays period-doubling, chaotic behaviour, and periodic windows.

3 Dragon-kings death

Dragon-kings are extreme events that belong to a special class of events. The extreme events are characterised by power law statistics. However, the dragon-kings are beyond the power laws. They exhibit humps in the tail of the distributions for the extended size of events [7].

We calculate the probability density function (PDF) of the temporal evolution of $|A_{2}|$. Figure 4 shows the appearance of dragon-kings in the three-wave interactions model. In Figs. 4(a) and 4(b), we consider $\delta_{3}=2.2$ and $\nu=-14.5$, and $\delta_{3}=2$ and $\nu=-15$, respectively. The dragon-kings are represented by a lump in the PDF tail. The events with amplitude above $10$ are outliers and generate the dragon-kings.

With regard to the four-wave model, the fourth-wave interacts with the first and second waves. Lopes and Chian [26] used a small sinusoidal wave to control chaos in the nonlinear three-wave coupling. They demonstrated that a desired periodic orbit can be obtained applying a fourth wave with small amplitude. Aiming to suppress the dragon-kings, we apply a fourth wave ($|A_{4}|$) during a time interval $\Delta t=0.4$ and $r=0.5$ when $d|A_{1}|/dt>0$, as shown in Fig. 5(a). Figure 5(b) exhibits the temporal evolution of $|A_{4}|$. The energy of the first wave is decreased due to the application of the fourth wave (Fig. 5(c)). In Fig. 5(c), we see the behaviour of $|A_{1}|$ for $\delta_{3}=2.2$ and $\nu=-14.5$ when $r=0$ (red line) and $r=0.5$ (black line).

Figure 6(a) displays the PDF for $r=0$ (red line), $r=0.5$ (black line), and $\Delta t=0.4$. In our simulations, the fourth wave is able to produce a small alteration in the PDF. The PDF maintains the same slope, but the hump in the tail disappears, namely dragon-kings death. In Fig. 6(b), we see Lorentzian pulses not only in the nonlinear three-wave interactions (blue line), but also in the wave interactions (green line) where the dragon-kings are suppressed. Then, a fourth-wave kills the dragon-kings and does not destroy the Lorentzian pulses.

4 Conclusions

In conclusion, we study extreme events in a nonlinear three-wave interactions model. This model exhibits periodic and chaotic behaviour. In the chaotic region, we find Lorentzian pulses. The Lorentzian pulses appear in many physical systems and have been used to characterise chaotic dynamics. Extreme events are usually rare and the event size distributions follow a power law. Depending on the system parameters, our simulations show a power law dependence in the wave amplitude distribution, except for larger values of the amplitude. The distribution tail has humps, that is an evidence of dragon-kings extreme events. We consider an external perturbation in the three-wave interactions by means of a fourth wave. Desired periodic orbits can be achieved applying a wave with small amplitude. In this work, we show that a fourth wave applied during a short time can kill dragon-kings. Therefore, the fourth-wave can prevent catastrophic dragon-kings events in the three-wave interactions. We believe that in other systems there is a night king, i.e. an external perturbation, that is able to kill only the dragon-kings.

Acknowledgment

This work was made possible by support from the following Brazilian government agencies: Fundação Araucária, CNPq, CAPES, and FAPESP (2011/19296-1, 2015/50122-0, and 2015/07311-7).