Show that a pitchfork bifurcation occurs when $\mu = 0$ for the system $$\dot{x} = \mu x + xy + 3y^2$$ $$\dot{y} = -2y + x^2 + 2xy^2$$.
Attempt at a solution If $\dot{x} = 0$ then $$x = \frac{-3y}{\mu + y}.$$
The Jacobian is $$J(x,y) = \begin{pmatrix}\mu + y & 6y + x\\ 2x + 2y^2 & -2 + 4xy\end{pmatrix}$$
I know that $\det J(0, 0) = -2\mu$ so the $(0, 0)$ fixed point passed from unstable to stable as we pass $\mu = 0$, so we probably have a subcritical pitchfork.
Related question: What would you look for in a transcritical or saddle node bifurcation? What's the general procedure for determining the bifurcation type in a 2D system?