Chapter 12.6, Problem 5E

In Problems $5-12$, either by hand or using a computer software package, sketch the phase plane diagrams for the given system. Discuss the stability of each limit cycle and critical point. Here $x=r\cos \theta$ and $y=r\sin \theta$.

$\frac{dx}{dt}=x-y+x\left(r^3-4r^2+5r-3\right)$

$\frac{dy}{dt}=x+y+y\left(r^3-4r^2+5r-3\right)$