 Rational equilibrium curve at a degenerate pitchfork

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 $r-(1+q)=k{u}^2+\cdots$. 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.