Abstract
The effect of adding the uncatalysed reaction step to the cubic autocatalator in a closed system is examined under the pooled chemical approximation. The addition of this extra step has a dramatic effect on the overall reaction over the parameter range where, without this step, the pooled chemical approximation breaks down. It is found that, no matter how slow the uncatalysed reaction, the pooled chemical approximation now remains valid throughout. There is a parameter range where there is oscillatory behaviour, these oscillations arising from points of Hopf bifurcation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Gray and S.K. Scott: Autocatalytic reactions in the isothermal continuous stirred tank reactor: isolas and other forms of multistability. Chem. Engng. Sci. 38 (1983) 29–43.
P. Gray and S.K. Scott: Autocatalytic reactions in the continuous stirred tank reactor: oscillations and instabilities in the system A + 2B → 3B; B → C. Chem. Engng. Sci. 39 (1984) 1087–1097.
P. Gray and S.K. Scott: Sustained oscillations and other exotic patterns of behaviour in isothermal reactions. J. Phys. Chem. 89 (1985) 22–32.
S.K. Scott: Reversible autocatalytic reactions in an isothermal CSTR. Chem. Engng. Sci. 38 (1983) 1701–1708.
A. D'Anna, P.G. Lignola and S.K. Scott: The application of singularity theory to isothermal autocatalytic open systems: the elementary scheme A + mB → (m + 1)B. Proc. Roy. Soc. A403 (1986) 341–363.
J.H. Merkin, D.J. Needham and S.K. Scott: Oscillatory chemical reactions in closed vessels. Proc. Roy. Soc. A406 (1986) 299–323.
J.H. Merkin, D.J. Needham and S.K. Scott: On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme. To be published in SIAM J. on Applied Mathematics.
B.F. Gray, S.K. Scott and P. Gray: Multiplicity for isothermal autocatalytic reactions in open systems: the influence of reversibility and detailed balance. J. Chem. Soc. Faraday Trans. 1 80 (1984) 3409–3417.
B.F. Gray and S.K. Scott: Multistability and sustained oscillations in isothermal open system. J. Chem. Soc. Faraday Trans. 1 81 (1985) 1563–1567.
J. Tyson and S. Kauffman: Control of mitosis by a continuous biochemical oscillation: synchronisation; spatially inhomogeneous oscillations. J. Math. Biology 1 (1975) 289–310.
M. Ashkenazi and H.G. Othmer: Spatial patterns in coupled biochemical oscillators. J. Math. Biology 5 (1978) 305–350.
M. Golubitsky and W.F. Langford: Classification of degenerate Hopf bifurcations. J. Diff. Equations 41 (1981) 375–415.
D.W. Jordan and P. Smith: Nonlinear ordinary differential equations. Oxford University Press (1983).
J.H. Merkin, D.J. Needham and S.K. Scott: A simple model for sustained oscillations in isothermal branched-chain or autocatalytic reactions in a well stirred open system, II: limit cycles and nonstationary states. Proc. Roy. Soc. A398 (1985) 101–116.
B.D. Hassard, N.D. Kazarinoff and Y.-H. Wan: Theory and applications of Hopf bifurcation, (London Mathematical Society Lecture Notes Series No. 41, edited by I.M. James), Cambridge University Press (1981).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Merkin, J.H., Needham, D.J. & Scott, S.K. On the structural stability of a simple pooled chemical system. J Eng Math 21, 115–127 (1987). https://doi.org/10.1007/BF00127669
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00127669