The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reaction-diffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full nonsmooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail. Abstract The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reactiondiffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full non-smooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail.
In the one-dimensional reaction-diffusion domain of this study, semi-analytical solutions are used for a delayed viral infection system with logistic growth. Through an ordinary differential equations system, the Galerkin technique is believed to estimate the prevailing partial differential equations. In addition, Hopf bifurcation maps are constructed. The effect of diffusion coefficient stricture and delay on the model is comprehensively investigated, and the outcomes demonstrate that diffusion and delay can stabilize or destabilize the system. We found that, as the delay parameter values rise, the values of the Hopf bifurcations for growth and the rates of viral death are augmented, whereas the rate of production is decreased. For the growth, production, and death rates strictures, there is determination of an asymptotically unstable region and a stable region. Illustrations of the unstable and stable limit cycles, as well as the Hopf bifurcation points, are found to prove the formerly revealed outcomes in the Hopf bifurcation map. The results of the semi-analytical solutions and numerical assessments revealed that the semi-analytical solutions are highly effective.
In this work, the semi-analytical solution is studied for the diffusive logistic equation with both mixed instantaneous and delayed density. The domain of reaction-diffusion in one dimension is shown. Delay partial differential equation is approximated with a delay ordinary differential equation system by using the Galerkin technique method. Steady-state solutions and stability analysis as well as bifurcation diagrams are derived. The effect of diffusion parameter and delay values is comprehensively studied; as a result, both parameters can destabilize or stabilize the model. We obtained that the decrease in values of the Hopf bifurcations for growth rate is associated with an increase in delay values, whereas the diffusion parameter is increased. Furthermore, comparisons between the numerical simulations and semi-analytical results present a good agreement for all examples and figures of the Hopf bifurcations. Examples of limit cycle and phase-plane map are plotted to confirm the benefits and accuracy of semi-analytical solutions result. For periodic solutions, an asymptotic method is studied after the Hopf bifurcation point for both one-and two-term semi-analytical systems.
Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.
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