= Rational equilibrium curve at a degenerate pitchfork
{title2=$r(u)=(1+u)(1+q/(1+ku)),\quad u=a^2$}
For $k>1$, the rational <equilibrium point> curve in the display has expansion $r=1+q+[1-q(k-1)]u+qk(k-1)u^2+\cdots$. Its cubic <pitchfork bifurcation> coefficient vanishes at $q=1/(k-1)$, leaving $r-(1+q)=ku^2+\cdots$. For larger $q$, the nonzero branch folds at $u=[\sqrt{q(k-1)}-1]/k$ and $r=[k-1+q+2\sqrt{q(k-1)}]/k$. If the other <eigenvalues> remain stable, the two folds bound a bistable wedge between stable zero and stable nonzero states, separated by an unstable branch. If another <eigenvalue> is critical, a higher-dimensional <centre manifold> is required instead.
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