On the structural stability of a simple pooled chemical system

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.

[1]  Martin Golubitsky,et al.  Classification and Unfoldings of Degenerate Hopf Bifurcations , 1981 .

[2]  S. Scott,et al.  A simple model for sustained oscillations in isothermal branched-chain or autocatalytic reactions in a well stirred open system - II. Limit cycles and non-stationary states , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[3]  S. Kauffman,et al.  Control of mitosis by a continuous biochemical oscillation: Synchronization; spatially inhomogeneous oscillations , 1975 .

[4]  S. Scott,et al.  On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme , 1987 .

[5]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[6]  S. Scott,et al.  The application of singularity theory to isothermal autocatalytic open systems: the elementary scheme A + mB = (m + 1)B , 1986, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[7]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C , 1984 .

[8]  Hans G. Othmer,et al.  Spatial patterns in coupled biochemical oscillators , 1977 .

[9]  S. Scott.  Reversible autocatalytic reactions in an isothermal CSTR: Multiplicity, stability and relaxation times , 1983 .

[10]  S. Scott,et al.  Oscillatory chemical reactions in closed vessels , 1986, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[11]  P. Gray,et al.  Sustained oscillations and other exotic patterns of behavior in isothermal reactions , 1985 .

[12]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability , 1983 .

[13]  D. Jordan,et al.  Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers , 1979 .