Qualitative analysis and Hopf bifurcation of a generalized Lengyel–Epstein model

Chen, Mengxin

doi:10.1007/s10910-022-01418-8

Cited by 3 publications

(2 citation statements)

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“…Asheghi [1] given the influence of the diffusion coefficient of the homogeneous steady state and obtained the direction of the Hopf bifurcation by investigating a reduced Gierer-Meinhardt model. Chen and Wang [2] reported the boundedness of the solution and studied the properties of the Hopf bifurcation for a generalized Lengyel-Epstein model. Muntari and Sengül [3] performed a rigorous characterization of the types and structure of the dynamic transitions and showed the relation between the dynamic transitions and the pattern formations of a Brusselator model in a 2D rectangular box.…”

Section: Introductionmentioning

confidence: 99%

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Hopf Bifurcation and Self-Organization Pattern of a Modified Brusselator Model

Chen¹

2023

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This paper reports the Hopf bifurcation and self-organization pattern of a modified Brusselator model. The model is a non-standard Brusselator model, it involves the nonlinear restraint term. For the non-diffusive model, we give the types of unique positive equilibrium. It is found that the unique positive equilibrium may be focus, node, or center and we establish their stability, respectively. Especially, there exists the spatial homogeneous Hopf bifurcation when the equilibrium is a center. The first Lyapunov number technique is applied to perform the direction of the spatial homogeneous Hopf bifurcation. In the sequel, the occurrence conditions of the Turing instability and the spatial inhomogeneous Hopf bifurcation are given for the diffusive model. Moreover, by using the normal form theory, we show that the Hopf bifurcation is supercritical or subcritical. Finally, the self-organization patterns induced by the Turing instability and periodic solutions resulting from the Hopf bifurcation are displayed by employing numerical simulations. Our theoretical predictions and numerical results reveal that the modified Brusselator model enjoys the temporal period oscillation and spatial oscillation due to the Hopf bifurcation and Turing instability, respectively. These results may help us to figure out the spatio-temporal dynamics of such modified Brusselator model.

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Section: Introductionmentioning

confidence: 99%

“…Note that the concentration of reactants will affect the reaction process and the production of reaction products. Thereby, if we suppose the autocatalytic step is a shorter term, namely, one replaces the term u 2 v by nonlinear restraint term u 2 v 1+u in the model (2), this gives the following modified Brusselator model…”

Section: Introductionmentioning

confidence: 99%