Chapter 7.5, Problem 16P

Consider the system of equations

$\begin{array}{l}x\text{'}=\mu x+y-x\left(x^2+y^2\right),\\ y\text{'}=-x+\mu y-y\left(x^2+y^2\right),\end{array}$ (i)

where $\mu$ is a parameter of unspecified sign. Observe that this system is the same as the one in Example 1, except for the introduction of $\mu$.

(a) Show that the origin is the only critical point.

(b) Find the linear system that approximates Eqs. (i) near the origin and find its eigenvalues. Determine the type and stability of the critical point at the origin. How does this classification depend on $\mu$?

(c) Referring to Example 1 if necessary, rewrite Eqs. (i) in polar coordinates.

(d) Show that when $\mu >0$, there is a periodic solution $r=\sqrt{\mu }$. By solving the system found in part (c), or by plotting numerically computed solutions, conclude that this periodic solution attracts all other nonzero solutions.

Note: As the parameter $\mu$ increases through the value zero, the previously asymptotically stable critical point at the origin loses its stability, and simultaneously a new asymptotically stable solution (the limit cycle) emerges. Thus the point $\mu =0$ is a bifurcation point; this type of bifurcation is called a Hopf bifurcation.