(i) For the dynamical system

$$\dot{x}=-x\left(x^{2}-2 \mu\right)\left(x^{2}-\mu+a\right),$$

sketch the bifurcation diagram in the $(\mu, x)$ plane for the three cases $a<0, a=0$ and $a>0$. Describe the bifurcation points that occur in each case.

(ii) For the case when $a<0$ only, confirm the types of bifurcation by finding the system to leading order near each of the bifurcations.

(iii) Explore the structural stability of these bifurcations by adding a small positive constant $\epsilon$ to the right-hand side of $(\uparrow)$ and by sketching the bifurcation diagrams, for the three cases $a<0, a=0$ and $a>0$. Which of the original bifurcations are structurally stable?