Consider the two-dimensional system $\displaystyle x'=Ax-r^2x$ where $\displaystyle r=\parallel x\parallel$ and A is a 2x2 constant real matrix with complex eigenvalues $\displaystyle \alpha\pm i\omega$. Prove that there exists at least one limit cycle for $\displaystyle \alpha>0$ and that there are none for $\displaystyle \alpha<0$.