# Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay

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## Abstract

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## 1. Introduction

## 2. Reaction-Diffusion Model and Spot Replication

## 3. Extreme Events Induced by Delayed Feedback

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Examples of 2D stationary localized structures obtained from numerical simulations of the model Equation 1. (

**a**) and (

**c**) correspond to a single spot in the spatial profile of the chemical concentrations X and Y, respectively. (

**b**) and (

**d**) correspond to four spots in the spatial profile of the chemical concentrations X and Y, respectively. Parameters are $A=0.6$, $B=0.65$, and $D=150$. The mesh integration is $256\times 256$ points.

**Figure 2.**Self-replicating spots obtained from numerical simulations of the model Equation 1. (

**a**)–(

**h**): time evolution of the chemical concentration X. (

**i**)–(

**p**): time evolution of the chemical concentration Y. Parameters are $A=0.6$, $B=0.75$, and $D=150$. The mesh integration is $256\times 256$ points.

**Figure 3.**The threshold associated with the pattern formation instability as a function of the strength of the delayed feedback, with $A=0.6$.

**Figure 4.**The wavelength of the Turing–Prigogine instability as a function of the strength of the delayed feedback, with $A=0.6$.

**Figure 5.**(

**a**) Space-time map showing the evolution of $X(x,t)$ in the Brusselator reaction-diffusion model. Triangles indicate pulses with an intensity 5–10-times larger than the stationary localized structures without delayed feedback. The red square shows an extreme event with amplitude $X>20$. The Brusselator parameters are $D=150$, $A=0.6$, $B=0.75$, and the feedback parameters are ${\eta}_{x}=-{\eta}_{y}=2.5$ and $\tau =10$. (

**b**) A cross-section at t = 78 showing the spatial profile $X\left(x\right)$ of the rogue wave.

**Figure 6.**The number of events as a function of the amplitude of the pulses in the semi-logarithmic scale. The parameters are the same as in Figure 5. The SWH denotes the significant wave height. The dashed line indicates 2 × SWH.

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**MDPI and ACS Style**

Tlidi, M.; Gandica, Y.; Sonnino, G.; Averlant, E.; Panajotov, K.
Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay. *Entropy* **2016**, *18*, 64.
https://doi.org/10.3390/e18030064

**AMA Style**

Tlidi M, Gandica Y, Sonnino G, Averlant E, Panajotov K.
Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay. *Entropy*. 2016; 18(3):64.
https://doi.org/10.3390/e18030064

**Chicago/Turabian Style**

Tlidi, Mustapha, Yerali Gandica, Giorgio Sonnino, Etienne Averlant, and Krassimir Panajotov.
2016. "Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay" *Entropy* 18, no. 3: 64.
https://doi.org/10.3390/e18030064