Bifurcation in the Lengyel–Epstein system for the coupled reactors with diffusion

Author

1 King Khalid University, Faculty of Science, Department of Mathematics, Abha 9004, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

10.21608/joems.2016.386762

Abstract

The main goal of this paper is to continue the investigations of the important system of
Fengqi et al. (2008). The occurrence of Turing and Hopf bifurcations in small homogeneous arrays
of two coupled reactors via diffusion-linked mass transfer which described by a system of ordinary
differential equations is considered. I study the conditions of the existence as well as stability
properties of the equilibrium solutions and derive the precise conditions on the parameters to show
that the Hopf bifurcation occurs. Analytically I show that a diffusion driven instability occurs at a
certain critical value, when the system undergoes a Turing bifurcation, patterns emerge. The
spatially homogeneous equilibrium loses its stability and two new spatially non-constant stable
equilibria emerge which are asymptotically stable. Numerically, at a certain critical value of diffusion
the periodic solution gets destabilized and two new spatially nonconstant periodic solutions
arise by Turing bifurcation.

Keywords

Journal of the Egyptian Mathematical Society
Volume 24, Issue 1
2016
Pages 25-29

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