In class we have studied $1D$ saddle, transcritical and pitchfork bifurcations. We did this by analysing their normal forms. For some parameter $\mu$, I have that the normal forms are:
Saddle Node: $\dot{x} = \mu - x^2$, Transcritical: $\dot{x} = \mu x - x^2$, Pitchfork: $\dot{x} = \mu x \pm x^3$.
The teacher explained that all dynamical systems which experience one of these bifurcations can be reduced to these normal forms. However, he only gave us the general example for the saddle node bifurcation for a generic dynamical system $\dot{x} = F(x;\mu)$.
He gave some geometric reasoning as to why for a saddle node bifurcation occurring at $\mathbf{\mu^{\ast}} = (x^{\ast},\mu^{\ast})$, we require:
$\partial_x F|_{\mathbf{\mu^{\ast}}} = 0$,
$\partial_{\mu} F|_{\mathbf{\mu^{\ast}}} \neq 0$,
$\partial_{xx} F|_{\mathbf{\mu^{\ast}}} \neq 0$.
This then results in the general form $F(x;\mu) = k\mu +lx^2 + \alpha \mu x + \beta \mu^2 + \cdots$ for which we can then allow $k=1$ and $l = -1$ in order to obtain the general form $F(x;\mu) = \mu - x^2 + \alpha \mu x + \beta \mu^2 + \cdots$. He then did the usual reduction steps in order to obtain the normal form.
Would someone mind showing me the same process but for the transcritical and pitchfork bifurcations?
Thanks