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Delay Effects on the Dynamics of the Lengyel–Epstein Reaction-Diffusion Model

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Mathematical Modeling and Applications in Nonlinear Dynamics

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 14))

Abstract

We investigate bifurcations of the Lengyel–Epstein reaction-diffusion model involving time delay under the Neumann boundary conditions. We first give stability and Hopf bifurcation analysis of the ordinary differential equation (ODE) models, including delay associated with this model. Later, we extend this analysis to the partial differential equation (PDE) model. We determine conditions on parameters of both models to have Hopf bifurcations. Bifurcation analysis for both models show that Hopf bifurcations occur by regarding the delay parameter as a bifurcation parameter. Using the normal form theory and the center manifold reduction for partial functional differential equations, we also determine the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions for the PDE model. Finally, we perform some numerical simulations to support analytical results obtained for the ODE models.

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Authors and Affiliations

  1. Department of Mathematics, TOBB University of Economics and Technology, Söğütözü Cad. No 43. 06560, Ankara, Turkey

    Hüseyin Merdan

  2. Çankaya University, Eskişehir Yolu 29.km, Yukarıyurtçu Mahallesi Mimar Sinan Caddesi No 4. 06790, Etimesgut, Ankara, Turkey

    Şeyma Kayan

Authors
  1. Hüseyin Merdan
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  2. Şeyma Kayan
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Correspondence to Hüseyin Merdan .

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Editors and Affiliations

  1. Dept of Mech.Eng & Industrial Engg., Southern Illinois Univ, Edwardsville, Edwardsville, Illinois, USA

    Albert C.J. Luo

  2. TOBB Univ. of Economics and Technology, Department of Mathematics, Ankara, Turkey

    Hüseyin Merdan

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Merdan, H., Kayan, Ş. (2016). Delay Effects on the Dynamics of the Lengyel–Epstein Reaction-Diffusion Model. In: Luo, A., Merdan, H. (eds) Mathematical Modeling and Applications in Nonlinear Dynamics. Nonlinear Systems and Complexity, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-26630-5_6

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  • DOI: https://doi.org/10.1007/978-3-319-26630-5_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26628-2

  • Online ISBN: 978-3-319-26630-5

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