Skip to main content
SpringerLink
Log in

Bautin bifurcation for the Lengyel–Epstein system

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

In this paper, complete analysis is presented to study Bautin bifurcation for the Lengyel–Epstein System

$$\begin{aligned} \left\{ \begin{array}{l} \frac{du}{dt}=a-u-\frac{4uv}{1+u^2},\\ \frac{dv}{dt}=\sigma b\left( u-\frac{uv}{1+u^2}\right) . \end{array}\right. \end{aligned}$$

Sufficient conditions for \(a\) and \(b\) are given for the system to demonstrate Bautin bifurcation. By using \(b\) and \(\alpha =a/5\) as bifurcation parameters and computing the first and second Lyapunov coefficients and performing nonlinear transformation, the normal form with unfolding parameters is derived to obtain the bifurcation diagrams such as Hopf and double limit cycle bifurcations. An example is given to confirm that the system has two limit cycles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953–2956 (1990)

    Article  CAS  Google Scholar 

  2. I.R. Epstein, I. Lengyel, Turing structures progress toward a room temperature, closed system. Phys. D Nonlinear Phenom. 84, 111 (1995)

    Article  Google Scholar 

  3. O. Jensen, V.O. Pannbacker, E. Mosekilde, G. Dewel, P. Borckmans, Turing structures and turbulence in a chemical reaction-diffusion system. Open Syst. Inf. Dyn. 3, 215–235 (1995)

    Article  Google Scholar 

  4. Y. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edn. (Springer, New York, 2004)

    Book  Google Scholar 

  5. I. Lengyel, I.R. Epstein, Global bifurcation and structure of Turing patterns in the 1-D Lengyel–Epstein model. Sciences 251, 650–652 (1991)

    Article  CAS  Google Scholar 

  6. I. Lengyel, I.R. Epstein, A chemical approach to designing Turing patterns in reaction-diffusion system. Proc. Natl. Acad. Sci. USA 89, 3977–3979 (1992)

    Article  CAS  Google Scholar 

  7. I. Lengyel, G. Ribai, I.R. Epstein, Experimental and modeling study of oscillations in the chlorine dioxide-iodine-malonic acid reaction. J. Am. Chem. Soc. 112(25), 9104–9110 (1990)

    Article  CAS  Google Scholar 

  8. N. Li, L. Shi, X. Wang, F. Guo, C. Yan, Experimental study of closed system in the chlorine dioxide-iodide-sulfuric acid reaction by UV–Vis spectrophotometric method. Int. J. Anal. Chem. (2011). doi:10.1155/2011/130102

  9. D.G. Míguez, S. Alonso, A.P. Munũzuri, F. Sagués, Experimental evidence of localized oscillations in the photosensitive chlorine dioxide-iodine-malonic acid reaction. Phys. Rev. Lett. 97, 178301 (2006)

    Article  Google Scholar 

  10. W. Ni, M. Tang, Turing patterns in the Lengyel–Epstein system for the CIMA reaction. Trans. Am. Math. Soc 357, 3953–3969 (2005)

    Article  Google Scholar 

  11. Q. Ouyang, V. Castets, J. Boissonade, J.C. Roux, P. De Kepper, Sustained patterns in chlorite-iodide reactions in a one-dimensional reactor. J. Chem. Phys. 95, 351–360 (1991)

    Article  CAS  Google Scholar 

  12. F. Yi, J. Wei, J. Shi, Diffusion-driven instability and bifurcation in the Lengyel–Epstein system. Nonlinear Anal. RWA 9, 1038–1051 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Computer and Information Sciences, Mississippi Valley State University, Itta Bena, MS, 38941, USA

    Xiaoqin P. Wu

  2. Department of Natural Sciences and Environmental Health, Mississippi Valley State University, Itta Bena, MS, 38941, USA

    Matthewos Eshete

Authors
  1. Xiaoqin P. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthewos Eshete
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoqin P. Wu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, X.P., Eshete, M. Bautin bifurcation for the Lengyel–Epstein system. J Math Chem 52, 2570–2580 (2014). https://doi.org/10.1007/s10910-014-0401-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-014-0401-6

Keywords

Mathematics Subject Classification

Search

Navigation