On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme.

*(English)*Zbl 0627.92026The dimensionless kinetic differential equation
\[
\dot x=\mu -xy^ 2,\quad \dot y=xy^ 2-y,
\]
of the chemical reaction \(P\to X\), \(X+2Y\to 3Y\), \(Y\to C\) is investigated. It is shown that there is one finite equilibrium point, with a Hopf bifurcation occurring at \(\mu =1\). The global phase portrait is also constructed for all positive values of \(\mu\). (To do so, the transformation \(u:=y/x\), \(v:=-1/x\) turns out to be useful.) From this it emerges that the stable limit cycle created at \(\mu =1\) by a Hopf bifurcation is destroyed at \(\mu_ 0\) \((<1)\) by an infinite period bifurcaton, due to the formation of a heteroclinic orbit by the separatrices from the equiibrium 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. The chemical reaction above has earlier been proposed by P. Gray and Scott [Chem. Engrg. Sci. 38, 29-43 (1983)] and may be considered as the core of almost all oscillatory chemical reactions.

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. The chemical reaction above has earlier been proposed by P. Gray and Scott [Chem. Engrg. Sci. 38, 29-43 (1983)] and may be considered as the core of almost all oscillatory chemical reactions.

Reviewer: J.Tóth

##### MSC:

92Exx | Chemistry |

80A30 | Chemical kinetics in thermodynamics and heat transfer |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |