Minimum models of damped and limit cycle oscillations in a polymerization

Abstract

A simple polymerization scheme { X → R 1 R j + X → R j + 1 ( j = 1 , 2 , … , ∞ ) R i + R j → p o l y m e r ( i = 1 , 2 , … ∞ ; j = 1 , 2 , … , ∞ ) has been studied introducing small modifications leading to a stable focus type steady state (with damped oscillations) or unstable focus type (which combined with a no return enclosure for phase trajectories will show cycle limit sustained oscillations). Two variables have been employed in this analysis: X∝ monomer, Y α ∑ j = 1 ∞ R j = radicals . Limit cycle oscillations requires the addition of autocatalysis with respect to the monomer, J+ X→2 X, and so does an “enzymatic” block { U + X → V V → U assuming that U ̇ = 0 . The combination of both collateral additions makes the steady state an unstable focus and allows a simple Poincaré–Bendixson proof for the existence of the limit cycle.