Simple models of oscillating polymerizations:Dimeric form autocatalysis

For two variable chemical oscillators, a simple strategy is to combine autocatalysis with a term including a binomial Michaelian denominator (constant+concentration). This strategy is applied in this paper to the context of polymerization reactions. In order to achieve autocatalysis a kinetic scheme of the form {A+M⟹λDD⟹δ2M is applied, where the dimeric form D is included. The Michaelian term can be obtained in different ways, for example: (i) a third compound Z flows through the system and eliminates the monomer, (ii) compound Z can be radical initiators, and (iii) compound Z, which eliminates monomer, can be a special conformation (adsorbent) of the polymer. Three mechanisms are developed using these Michaelian terms. After decreasing the number of variables to two, by assuming a non-significant variation of the others, and normalizing the kinetic equations with respect to a certain steady state, phase plane techniques are applied to prove the existence of a limit cycle in the three models. Both monomer concentration and propagator concentration in the three models (cases 2.1 and 2.2) or the concentration of the adsorbent polymers are taken as oscillating variables.