Skip to main content
SpringerLink
Log in

The reversible LVA model

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

Abstract

We show that the solutions of the reversible LVA model are bounded. We give a sufficient condition for the existence of a globally asymptotically stable stationary point. In the case where only the first reaction is reversible in the LVA model, we use Liapunov functions to investigate the global behaviour of the system. A certain parameterL plays an important role in the phase portrait. The stationary point in the axis x is an attractor forL > 1, with a basin containing the open positive quadrant. ForL < 1 there exists a unique positive stationary point, which is stable forL > 1/2 and loses its stability forL < 1/2 via supercritical Hopf bifurcation.

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

Similar content being viewed by others

References

  1. Z. Noszticzius, H. Farkas and Z.A. Schelly, Explodator: A new skeleton mechanism for the halate driven chemical oscillators, I. Chem. Phys. 80 (1984)6062–6070.

    Google Scholar 

  2. H. Farkas and Z. Noszticzius, Generalized Lotka-Volterra schemes. Construction of two-dimensional explodator cores and their Liapunov functions via “critical” Hopf bifurcation, I. Chem. Soc. Faraday Trans. 2 81 (1985)1487–1505.

    Google Scholar 

  3. A. Dancsó, H. Farkas, M. Farkas and Gy. Szabó, Investigations on a class of generalized LotkaVolterra schemes, Acta Appl. Math. 23 (1991)103–127.

    Google Scholar 

  4. H. Farkas, Z. Noszticzius, C.R. Savage and Z.A. Schelly, Two-dimensional explodators. 2: Global analysis of the Lotka-Volterra-Brusselator (LVB) model, Acta Phys. Hung. 66 (1989)203–220.

    Google Scholar 

  5. H. Farkas and Z. Noszticzius, Explosive, conservative and dissipative systems and chemical oscillators, in:Advances in Thermodynamics, Vol. 6, ed. S. Sieniutycz and P. Salamon, pp. 299–335.

  6. P. Gray and S.K. Scott, J. Phys. Chem. 87 (1983)1835.

    Google Scholar 

  7. R.H. Hering, Oscillations in Lotka-Volterra systems of chemical reactions, J. Math. Chem. 5 (1990)197–202.

    Google Scholar 

  8. S. Lengyel, J. Chem. Phys. 88 (1988)1617–1621.

    Google Scholar 

  9. S. Lengyel and I. Gyarmati, J. Chem. Phys. 75 (1981)2384–2390.

    Google Scholar 

  10. I.R. Shafarevich,Basic Algebraic Geometry (Springer, 1974).

  11. J.J. Tyson,The Belousiv-Zhabotinskii Reaction (Springer, 1976).

  12. J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, 1983).

Download references

Author information

Authors and Affiliations

  1. Institute of Physics, Technical University of Budapest, H-1521, Budapest, Hungary

    Peter L. SIMON

Authors
  1. Peter L. SIMON
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

SIMON, P.L. The reversible LVA model. J Math Chem 9, 307–322 (1992). https://doi.org/10.1007/BF01166095

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01166095

Keywords

Search

Navigation