I have the following dynamical system $ \dot{x} = \mu - mx - xy^2 $ $ \dot{y} = -\mu y + xy^2 $ and I need to "analyse" any bifurcations that occur as $m$ and $\mu$ are varied. I've worked out that the critical points are $(\frac{\mu}{m}, 0)$ for $m \neq 0$, $(\frac{2\mu}{1 \pm \sqrt{1 - 4m}}, \frac{1 \pm \sqrt{1 - 4m}}{2})$ for $m \leq \frac{1}{4}$, and the entire $y$ axis for $\mu = 0$. To find the bifurcations, I worked out when the critical points were non-hyperbolic by finding the determinant and trace of the Jacobian matrix at each critical point, and setting them to zero. It seems that there is a saddle-node bifurcation occurring at $m = \frac{1}{4}$ for the points $(\frac{2\mu}{1 \pm \sqrt{1 - 4m}}, \frac{1 \pm \sqrt{1 - 4m}}{2})$, and a Hopf bifurcation occurring at $\mu = \frac{1 + \sqrt{1 - 4m}}{2} = y$ for the point $(\frac{2\mu}{1 + \sqrt{1 - 4m}}, \frac{1 + \sqrt{1 - 4m}}{2})$.
The problem is, I don't know how to prove that these bifurcations are in fact saddle-nodes and Hopfs, respectively. In a one-dimensional system, I know that one has to calculate certain partial derivatives of the original equation (with respect to the variable and parameter) to confirm that a bifurcation is a saddle-node. But since this is a two-dimensional system, I'm not sure what do to. Similarly, the fact that there are two parameters seems to complicate the process of confirming that the bifurcation is a Hopf. I have managed to get the system into normal form for each bifurcation. I did this by first shifting $x$, $y$, and $m$ so that the bifurcations in each case occurred at $(\bar{x}, \bar{y}, \bar{m}) = (0, 0, 0)$. Then I substituted $\bar{x}$, $\bar{y}$, and $\bar{m}$ (the shifted variables) into the original equations to get a new system of the form $\dot{\bar{\mathbf{x}}} = A\bar{\mathbf{x}} + \text{nonlinear terms},$ and I transformed this into normal form in the usual way (i.e. by defining $\mathbf{u} = P^{-1}\bar{\mathbf{x}}$, where $P$ is the matrix of eigenvectors of $A$, to get $\dot{\mathbf{u}} = \Lambda\mathbf{u}$).
Presumably, I have to use this new normal form system in order to classify the bifurcations that occur in each case, but I'm not sure how to do this.