# Feedback Control for a Diffusive and Delayed Brusselator Model: Semi-Analytical Solutions

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulations Model

## 3. The Galerkin Technique Methodology

## 4. Theoretical Framework for the Existence of Hopf Bifurcation

## 5. Stability and Hopf Bifurcation Analysis

#### 5.1. Hopf Bifurcation Areas

#### 5.2. Bifurcation Diagrams, Periodic Oscillation, and Phase-Plane Map

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(color online) (

**a**,

**b**) The Hopf bifurcation curves in the $\tau -A$ and $\tau -B$ maps. The line of red crosses refers to the results of the partial differential equation (PDE) numerical simulation and the black dots are the results of the two terms in the semi-analytical system. The positive values that were used are $D=0.01$, $H=0.1$, and $B=10$ in (

**a**) and $A=2$ in (

**b**).

**Figure 2.**(color online) (

**a**,

**b**) The Hopf bifurcation lines of the $H-B$ and $D-H$ planes. The analytical simulation results for the two terms (black solid line) and the numerical simulation of the PDEs (line of red crosses) were obtained. The parameters used were $D=0.01$, $\tau =1$, and $A=1$ for (

**a**); and $\tau =1$, $A=5$, and $H=0.01$ for (

**b**).

**Figure 3.**(color online) (

**a**,

**b**) The Hopf bifurcation maps in the $A-B$ diagrams versus examples of feedback parameter H and the diffusion parameter D, respectively. The two-term semi-analytical simulation is plotted. The parameter values applied here were $\tau =1$ and $D=0.01$ for (

**a**); and $\tau =1$ and $H=0.01$ for (

**b**).

**Figure 4.**(

**a**,

**b**) The Hopf bifurcation areas for diffusion coefficient value D against the parameters of feedback control H and delay value $\tau $, respectively. The semi-analytical outcomes of the two-term results are plotted. The parameter values used were $\tau =1$, $A=1$, and $B=10$ for (

**a**); in (

**b**), $A=1$, $H=0.01$, and $B=10$.

**Figure 5.**(color online) (

**a**) The Hopf bifurcation map in the $B-A$ plane and (

**b**) the frequency of the periodic results $\omega $ against B. Here, we can see the results for two cases: $H=0$, where there is no feedback control term; and the $H=0.1$ curve. The other parameter values are $D=1$ and $\tau =1$.

**Figure 6.**(color online) The steady-state concentration of the interacting chemical species of X and Y versus control B. The analytical results for one and two terms are shown as a black solid and a blue dashed curve, respectively. The red dotted points refer to the numerical simulation of the PDE model. The parameters used were $H=A=D=1$ and $\tau =0$.

**Figure 7.**(color online) (

**a**,

**b**) The bifurcation map of reactant concentrations X and Y versus control parameter concentration B. The analytical results for one and two terms are indicated by black solid and blue dashed curves, respectively. The PDE numerical simulation (dotted red curve) is also shown. The parameters used were $A=1$, $\tau =1$, $H=0.1$, and $D=0.01$.

**Figure 8.**(color online) (

**a**,

**b**) The bifurcation diagram of chemical concentrations X and Y against control concentration B for three different cases of feedback control parameter H: H = 0.05, 0.10, and 0.15. The analytical results of the two terms are plotted. The values of the parameters were $\tau =1$, $A=1$, and $D=0.01$.

**Figure 9.**(

**a**,

**b**) The reactant concentrations of the two interacting chemical species X and Y against time. The one-term analysis is shown as a blue dashed line; the black solid line indicates the analytical results of the two-term analysis. The numerical simulation of the PDE system (red dotted line) is plotted. The parameters used were $B=2$, $D=0.01$, $\tau =1$, $H=0.1$, and $A=1$ (color available online).

**Figure 10.**(color online) (

**a**,

**b**) The limit cycle maps for the reactant concentrations X and Y over time. The analytical results for one and two terms are shown as black solid and blue dashed curves, respectively, and the PDE numerical scheme (dotted red curve) is also presented. The parameters used were $B=3$, $D=0.01$, $\tau =1$, $H=0.1$, and $A=1$.

**Figure 11.**(color online) (

**a**) The phase planes in a 2D map for the concentration of reactant X against Y. The analytical results of the two-term (red dotted line) and one-term (blue dashed line) results are plotted, where the black solid line represents the numerical simulation scheme of the PDE results, with $B=3$, $D=0.01$, $H=0.1$$\tau =1$, and $A=1$. (

**b**) 2D phase-plane maps for the two-term analytical outcomes with four different examples of B: $B=3$ (solid black), $B=3.5$ (dotted red), $B=4$ (dotted blue), and B = $B=4.5$ (dotted black).

**Figure 12.**(color online) Limit cycle map for the two-term analytical results: $A=1$, $B=3$, and $\tau =1$ for three different examples of H and D, with $D=0.01$ for (

**a**), and $H=0.1$ for (

**b**).

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**MDPI and ACS Style**

Alfifi, H.Y.
Feedback Control for a Diffusive and Delayed Brusselator Model: Semi-Analytical Solutions. *Symmetry* **2021**, *13*, 725.
https://doi.org/10.3390/sym13040725

**AMA Style**

Alfifi HY.
Feedback Control for a Diffusive and Delayed Brusselator Model: Semi-Analytical Solutions. *Symmetry*. 2021; 13(4):725.
https://doi.org/10.3390/sym13040725

**Chicago/Turabian Style**

Alfifi, Hassan Yahya.
2021. "Feedback Control for a Diffusive and Delayed Brusselator Model: Semi-Analytical Solutions" *Symmetry* 13, no. 4: 725.
https://doi.org/10.3390/sym13040725