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

Abstract

This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition able to determine the Hopf bifurcation point is found. Full maps of the Hopf bifurcation regions for the interacting chemical species are shown and discussed, indicating that the time delay, feedback control, and diffusion parameters can play a significant and important role in the stability dynamics of the two concentration reactants in the system. As a result, these parameters can be changed to destabilize the model. The results show that the Hopf bifurcation points for chemical control increase as the feedback parameters increase, whereas the Hopf bifurcation points decrease when the diffusion parameters increase. Bifurcation diagrams with examples of periodic oscillation and phase-plane maps are provided to confirm all the outcomes calculated in the model. The benefits and accuracy of this work show that there is excellent agreement between the analytical results and numerical simulation scheme for all the figures and examples that are illustrated.

Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method

“…Thus, we can compute the approximated solution by evaluating it at any point. Other advantages of the analytical solutions, such as the long-term behavior of the solution, can be seen in [14,15,[18][19][20][21][22]. The errors of the LT solutions usually decrease as time increases.…”

Stability analysis and Hopf bifurcation for two-species reaction-diffusion-advection competition systems with delays

“…The Galerkin technique has been proven on several models, such as the viral infection model [7], Nicholson's blowfly model [14], the business cycle system [11], classes of delay logistic equations [8,12,13], the limited-food model [6], neural network model [9], Gray and Scott's system [48], and the Belousov-Zhabotinsky model [15]. Overall, the outcomes demonstrate strong agreement between analytical and numerical solutions.…”

“…The existence of time delays in reaction-diffusion systems are also significant, and changing the delays can radically change the behavior of a model, for instance replacing domain-wide stability to a spectrum of unstable results [4,10,21,24,53]. In this work, we consider, within heterogeneous advection environments, how time delays affect the non-linear, two-species competition in the following reaction-diffusion model:…”

Galerkin-Bernstein Approximations of the System of Time Dependent Nonlinear Parabolic PDEs

“…Because of its potential to provide a close analytical solution, the fractional-order Brusselator model was studied by Faiz et al [3]. The Brusselator system stability of a reaction-diffusion cell as well as the Hopf bifurcation analysis of the system have been detailed by Alfifi [4]. Qamar has analyzed the dynamics of the discrete-time Brusselator model with the help of the Euler forward and nonstandard difference schemes [5].…”