Asymptotic calculation of a limit cycle (Q1238236)

Consider the system \[ dx/dt=a-(b+1)x+x^2y,\, dy/dt=bx-x^2y \tag{*} \] modelling a biochemical reaction. For \(b>1+a^2\) there exists a stable limit cycle around the critical point \(x=a\), \(y=b/a\). Using singular perturbation techniques the author calculates in the case \(b \gg 1\) (relaxation oscillation) asymptotically the limit cycle. To this end he transforms (*) into an equation of order two, gives asymptotic expansions in distinct regions and matches this expansions. Finally, he compares the obtained results for fixed \(b\) and \(a\) with that ones calculated numerically by \textit{B. Lavendatal}. [J. Theor. Biol. 32, 283--292 (1971)].