Bifurcation points of differential equation (example) Assume the differential equation:
$$
x'=\lambda^2-8a\lambda x+2x^2, \quad a\in \mathbb{R}.
$$
The critical points are the solutions to the equation:
$$
x'=0 \iff 2x^2-8a\lambda x +\lambda^2=0\tag{1}
$$
which admits solutions:
$$
x=\lambda \cdot \frac{8a \pm \sqrt{64a^2-8}}{2}
$$
Now, having in mind that by bifurcation point, we mean a point where change of the number of equilibrium points occur, it seems to me that the number of $(1)$'s solutions depends only on $a$ and not on $\lambda$. Is this the case or did I get something wrong?

It appears to me our OP LoneBone is correct; I can find no errors in his/her logic.
