Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 60 / 1

Choose amplitude coordinates in which a quarter-turn and a diagonal reflection act as $r(A,B)=(B,-A)$ and $s(A,B)=(-A,B)$. These generate the natural two-dimensional group representation of the dihedral group $D_4$. An equivariant dynamical system satisfies $F(gv)=gF(v)$. The two independent sign changes make $F_A$ odd in $A$ and even in $B$, and $F_B$ odd in $B$ and even in $A$. The quarter-turn equates the two linear coefficients and the two sets of cubic coefficients. No quadratic monomial is allowed. Thus the square-symmetric cubic steady-state normal form isThe bifurcation parameter has been scaled so that the critical linear eigenvalue is $\mu$. Nondegeneracy at cubic order means $a\ne 0$, $a+b\ne 0$ and $a-b\ne 0$. The equilibrium equations factor as $A(\mu -a{A}^2-b{B}^2)=B(\mu -a{B}^2-b{A}^2)=0$. When both components are nonzero, subtracting the two bracketed expressions gives $(a-b)(A^2-B^2)=0$. Therefore the complete leading list of small steady branches is:

Each family is one orbit of the dihedral group. Each nonzero state retains one reflection fixing its amplitude line; its full spatial symmetry depends also on the chosen planforms. Higher-order terms perturb the branch amplitudes but preserve the nondegenerate local classification. For $\mu >0$, an axial branch is asymptotically stable precisely when $a>0$ and $b>a$, while a diagonal branch is asymptotically stable precisely when $a+b>0$ and $b<a$. Every cubic branch lying on $\mu <0$ has a positive radial eigenvalue $-2\mu$, so it is unstable. In the exceptional isotropic cubic case $a=b$, the cubic truncation instead has a circle of equilibria; higher-order anisotropy is then needed, which is why that case was excluded.

The third planform is fixed by a diagonal reflection and changes sign under a quarter-turn. Extend the preceding group representation byThe other diagonal reflection changes only the sign of $B$. Consequently the first two components of an equivariant dynamical system have factors $A$ and $B$, whereas its third component is even in each of $A,B$. The quarter-turn exchanges $A^2$ and $B^2$ while negating $C$. Its admissible quadratic coupling is therefore $A^2-B^2$ in the third equation and opposite $AC,BC$ couplings in the first two. At cubic order the only new terms are $A{C}^2,B{C}^2,C^3,C(A^2+B^2)$. The linear action on the two-dimensional irreducible mode has one coefficient ${\mu }_1$, while the distinct scalar representation has its own coefficient ${\mu }_2$. A small splitting of the two marginal linear modes supplies these two unfolding parameters. With real coefficients the resulting amplitude equations areThese equations follow by listing all equivariant monomials through degree three; the displayed minus signs are coefficient conventions, not consequences of symmetry.

In the specified reduced system put $A=B=s$, $C=0$. A nonzero branch exists for ${\mu }_1>0$, with $q=s^2={\mu }_1/(1+\lambda )$. Write perturbations as $u=(\delta A+\delta B)/2$, $v=(\delta A-\delta B)/2$, and $c=\delta C$. Linearization givesFor the square-pattern interaction with a sign-changing scalar mode, the remaining eigenvalues have sum $\tau =p+{\mu }_2$ and product $D=p{\mu }_2+4{\alpha }_1{\gamma }_{1}q=p({\mu }_2+K)$, where $K=2{\alpha }_1{\gamma }_1/(\lambda -1)>0$. A real two-dimensional stability matrix has both eigenvalues in the left half-plane exactly when its trace is negative and its determinant is positive. ThereforeIf ${\mu }_2<-K$, the block has negative determinant and one growing direction, explaining the instability at sufficiently small ${\mu }_2$. At the lower endpoint ${\mu }_2=-K$, one eigenvalue is zero and the other is $p-K<0$. The symmetry $(A,B,C)\mapsto (B,A,-C)$ fixes the diagonal state and negates its critical $(v,c)$ direction. Thus a generic nonlinear unfolding has a pitchfork bifurcation to unequal-amplitude states with nonzero $C$.

At the upper endpoint ${\mu }_2=-p$, the trace vanishes andThe symmetric eigenvalue remains negative. The critical conjugate pair crosses the imaginary axis with derivative $d\mathrm{Re}\sigma /d{\mu }_2=1/2$. If its cubic Hopf normal form coefficient is nonzero, this is a Hopf bifurcation and nearby periodic solutions exist on one side. Their stability and the side of existence depend on that coefficient; the linear calculation does not determine them. It does establish the generic possibility of oscillating square patterns. The equality $p=K$ is a double-zero degeneracy, outside this nondegenerate Hopf bifurcation argument.