On the Creation, Growth and Extinction of Oscillatory Solutions for a Simple Pooled Chemical Reaction Scheme

Abstract

The equations which govern a simple pooled chemical reaction scheme are analysed in detail in terms of a nondimensional parameter $\mu $, which represents the amount of the pooled chemical originally present. It is shown that there is one finite equilibrium point, with a Hopf bifurcation occurring at $\mu = 1$. The phase plane at infinity is then examined and it is shown that there are equilibrium points at infinity at the positive ends of both axes, the nature of which are discussed. This, together with a knowledge of the global phase portraits for $\mu \ll 1$ and $\mu \gg 1$, enables the global phase portrait to be constructed for all positive $\mu $. From this it emerges that the stable limit cycle created at $\mu = 1$ by a Hopf bifurcation is destroyed at $\mu _0 ( \mu _0 < 1 )$ by an infinite period bifurcation, due to the formation of a heteroclinic orbit by the separatrices from the equilibrium points at infinity. The form of this heteroclinic orbit is then discussed in detail, and in particular, it is shown that the value of $\mu _0 $ can be determined by simple numerical integration.