= Hopf coefficient of the Brusselator
{title2=$l_1=-(a+2)/8$}
For $u'=1-(b+1)u+au^2v$, $v'=bu-au^2v$, 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 $x'=-\sqrt a\,y+(1-a)x^2-2\sqrt a\,xy-ax^3-\sqrt a\,x^2y$, $y'=\sqrt a\,x$. 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 $\sqrt a$.
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