Minimum models of damped and limit cycle oscillations in a polymerization

A simple polymerization scheme{X→R1Rj+X→Rj+1(j=1,2,…,∞)Ri+Rj→polymer(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, Yα∑j=1∞Rj=radicals.Limit cycle oscillations requires the addition of autocatalysis with respect to the monomer, J+X→2X, and so does an “enzymatic” block{U+X→VV→Uassuming that U̇=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.