Consider the system [begin{aligned} x^{prime} & =-y+xleft[mu-x^{2}-y^{2} ight] y^{prime} & =x+yleft[mu-x^{2}-y^{2} ight] end{aligned}] Rewrite this system in polar form. Look at the behavior of

Consider the system

\[\begin{aligned} x^{\prime} & =-y+x\left[\mu-x^{2}-y^{2}\right] \\ y^{\prime} & =x+y\left[\mu-x^{2}-y^{2}\right] \end{aligned}\]

Rewrite this system in polar form. Look at the behavior of the \(r\) equation and construct a bifurcation diagram in \(\mu r\) space. What might this diagram look like in the three-dimensional \(\mu x y\) space? (Think about the symmetry in this problem.) This leads to what is called a Hopf bifurcation.