A nonlinear system is described by

\[
\begin{aligned}
& \dot{x}_{1}=x_{1}+x_{2}-\left(x_{1}^{3}+x_{1} x_{2}^{2}ight) \\
& \dot{x}_{2}=x_{2}-x_{1}-\left(x_{2}^{3}+x_{1} x_{2}^{2}ight)
\end{aligned}
\]

If \(r^{2}=x_{1}^{2}+x_{2}^{2}\), show that the system has a limit cycle at \(r=1\).