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it seems that I need some elements of bifurcation theory for my research, and I'm a bit puzzled at the moment by some basic stuff. I'm reading the beginning of the book 'Singularities and groups in bifurcation theory' from Golubitsky and Schaeffer.

Let's say we have an equation:

$f(x,\lambda) = -\lambda x + x^3 = 0$

this equation is the simplest form of the pitchfork bifurcation.

Now from what I read in the book of Golubitsky, for a general equation $g(x,\lambda)=0$, we can recognize a pitchfork bifurcation by the following conditions at $(x,\lambda)= (0,0)$:

$g=g_x=g_{xx}=g_{\lambda} = 0\mbox{ and } g_{xxx}\ g_{\lambda x} < 0.$

We can easily check that the function $f(x,\lambda) = -\lambda x + x^3$ verifies these conditions.

However, let us now do the simple change of variable $x = u+v$ and $\lambda = u-v$, this corresponds to a rotation of $\pi/2$ and yields the new equation

$f(x,y) = F(u,v) = -(u^2 -v^2) + (u+v)^3 = 0.$

Now if we check the conditions for a pitchfork perturbation we see for instance that $F_{uu}(0,0)=-2\neq 0$ and that $F_{uuu}(0,0)F_{uv}(0,0) = 0$. So this means that we don't have a pitchfork anymore. In fact I thought this would be considered a so-called simple bifurcation, but I might be wrong about that.

What I don't understand is that geometrically, the equations $f(x,y)=0$ and $F(u,v)=0$ lead to exactly the same curves, up to a rotation, so why would they be classified as different type of bifurcations?

The thing is, I am not interested in application of these results to study the stability of an ODE, my problem is purely geometrical, so for me this change of coordinate should not make any difference in my analysis.

Basically I have an equation of the type $f(x,y)=0$ with $\nabla f(0,0)=0$ and I want to know what the solution set looks like in the vicinity of $(0,0)$. I was hoping to get a classification of all the possible configurations (or at least the simplest ones) using bifurcation theory, using conditions on the higher derivatives of $f$, but then I got puzzled by the example I showed you.

Any idea where to find such a classification?

Multiverse
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    The point of bifurcation theory is that you have a family of differential equations indexed by a parameter. The role of the state $x$ and the parameter $\lambda$ is a priori distinct. We are interested in the change of the qualitative behaviour of the ODE as the $\lambda$ varies. At bifurcation points this change is abrupt in the sense that something qualitatively very different happens. Your change of variables mixes states and parameters. From the point of view of understanding differential equations this does not really make sense. You want to look at books on singularity theory proper. – Fabian Wirth Aug 02 '16 at 20:58
  • Ok, so the change of variable is meaningless for the study of ODEs, this makes sense. In fact, I found that I can use Morse's Lemma to solve my problem, which is purely geometrical. Now I am wondering if there is something like a Morse Lemma for degenerate critical points. – Multiverse Aug 21 '16 at 13:12

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