= Solution
A codimension-three degeneracy is possible when the double-zero <eigenvalue> condition and the vanishing cubic restoring coefficient of the <pitchfork bifurcation> coincide. Equating $q_D$ and $q_{TB}$ gives
$$
\boxed{k=\frac2{1+\sigma},\qquad
h=\frac{8\sigma^2}{1+\sigma+2\sigma^2},\qquad
\alpha^2=\frac{\pi^2(1+\sigma)}{2\sigma^2}.}
$$
Equivalently $L=\sqrt{2\sigma^2/(1+\sigma)}$. For every $0<\sigma<1$, this is a physically admissible aspect ratio and $0<h<4$. The other two parameters are $q=(1+\sigma)/(1-\sigma)$ and $r=2/(1-\sigma)$.
At this point the fifth-order restoring term is nonzero: the steady relation is $r-r_{TB}=ka^4+O(a^6)$ with $k>0$. It is therefore a degenerate reflection-symmetric <Bogdanov–Takens bifurcation> with three independent conditions: zero <determinant>, zero frequency, and zero cubic restoring coefficient. Varying the aspect ratio supplies the third parameter, in addition to $r,q$. At a fixed generic aspect ratio it is not a codimension-two event available by varying only $r,q$.
It is \b[not a triple-zero eigenvalue]: the three-mode characteristic polynomial still has the noncritical factor $s+1+2\sigma$, while the $c,e$ <eigenvalues> remain strictly negative. The extra degeneracy is nonlinear. A fifth-order, two-dimensional <normal form> is needed for its complete unfolding; the one-dimensional quintic <pitchfork bifurcation> used away from $q_{TB}$ cannot be substituted at the double-zero point.
Back to article page
