Bifurcation in the Lengyel–Epstein system for the coupled reactors with diffusion | ||
| Journal of the Egyptian Mathematical Society | ||
| Volume 24, Issue 1, 2016, Pages 25-29 PDF (2.31 M) | ||
| DOI: 10.1016/j.joems.2014.06.020 | ||
| Author | ||
| Shaban Aly1, 2 | ||
| 1King Khalid University, Faculty of Science, Department of Mathematics, Abha 9004, Saudi Arabia | ||
| 2Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt | ||
| 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 | ||
| Lengyel–Epstein; Coupled reactors; Hopf bifurcation; Turing instability | ||
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