State Dulac's Criterion and the Poincaré-Bendixson Theorem regarding the existence of periodic solutions to the dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{2}$. Hence show that

$$\begin{aligned} &\dot{x}=y \\ &\dot{y}=-x+y\left(\mu-2 x^{2}-y^{2}\right) \end{aligned}$$

has no periodic solutions if $\mu<0$ and at least one periodic solution if $\mu>0$.