Bifurcations with dihedral symmetry

Any function invariant under the Dn action on the plane is a function of the two fundamental homogeneous polynomials:
$$N=x^2+y^2 = |z|^2,\quad \mbox{and} \quad P = \mathrm{Re}(z^n)$$ The following diagrams show level sets of the functions $$h_\lambda(z) = \lambda N +aN^2 + bP.$$ So \(h_\lambda\) has a local min at the origin when \(\lambda\gt0\).

We need to assume \(b\neq0\), otherwise the function \(h_0\) does not have an isolated singularity.
And w.l.o.g, we can assume \(b>0\) (after a possible rotation in the plane).
(If \(b=0\) we would need to take into account higher order terms such as \(P^2\)).
Mostly we take \(a=b=1\), but for \(n=4\) we must take \(a\ne\pm b\).

Dn λ<0 λ>0 discussion
n=3 D3 D3 On both sides of the bifurcation, there are 3 saddle points; this is a traditional transcritical bifurcation
n=4 (i) D4 D4 (a<b) On both sides of the bifurcation, there are 4 saddle points; this is a traditional transcritical bifurcation
n=4 (ii) D4 D4 (a>b) On one side there are 4 saddles and 4 local minima, and on the other no bifurcating critical points
n=5 D4 D4 On one side there are 5 saddles and 5 local minima, and on the other no bifurcating critical points
n=6 D4 D4 On one side there are 6 saddles and 6 local minima, and on the other no bifurcating critical points
In the diagrams, if points \(p\) and \(q\) are such that \(p\) is coloured bluer than \(q\), then \(h(p) \lt h(q)\).

Remark: Notice how \(n = 3\) and \(n = 4(i)\) are similar, while \(n = 4(ii)\) and \(n \gt 4\) are similar.
The general pattern depends on which of the terms \(aN^2\) or \(bP\) dominates.
Notice that all the bifurcating critical points lie in lines of symmetry;
this means that the solutions they represent in a given problem will have an order 2 symmetry.

James Montaldi
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