 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 $q_{TB}$ givesEquivalently $L=\sqrt{2{\sigma }^2/(1+\sigma )}$. 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 $r-r_{TB}=k{a}^4+O(a^6)$ 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 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.