Any function invariant under the **D**_{n} 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 \(a\ne\pm b\).

D_{n} |
λ<0 | λ>0 | discussion |
---|---|---|---|

n=3 | On both sides of the bifurcation, there are 3 saddle points; this is a traditional transcritical bifurcation | ||

n=4 (i) | (a<b) On both sides of the bifurcation, there are 4 saddle points; this is a traditional transcritical bifurcation | ||

n=4 (ii) | (a>b) On one side there are 4 saddles and 4 local minima, and on the other no bifurcating critical points | ||

n=5 | On one side there are 5 saddles and 5 local minima, and on the other no bifurcating critical points | ||

n=6 | On one side there are 6 saddles and 6 local minima, and on the other no bifurcating critical points |

**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.

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