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?