 Hopf coefficient of the Brusselator

For $u^{\prime }=1-(b+1)u+a{u}^{2}v$, $v^{\prime }=bu-a{u}^{2}v$, the equilibrium $(1,b/a)$ has a Hopf bifurcation at $b=1+a$. Coordinates $x=u-1$, $y=-\sqrt{a}[(u-1)+(v-b/a)]$ give the threshold equations $x^{\prime }=-\sqrt{a}y+(1-a)x^2-2\sqrt{a}xy-a{x}^3-\sqrt{a}{x}^{2}y$, $y^{\prime }=\sqrt{a}x$. In the usual planar radial normalization the cubic coefficient is $-(a+2)/8<0$, proving a nondegenerate supercritical bifurcation. The leading small-amplitude angular frequency is $\sqrt{a}$.