2.II.8D

For all solutions of

$\begin{aligned} &\dot{x}=\frac{1}{2} \alpha x+y-2 y^{3} \\ &\dot{y}=-x \end{aligned}$

show that $d K / d t=\alpha x^{2}$ where

$K=K(x, y)=x^{2}+y^{2}-y^{4}$

In the case $\alpha=0$, analyse the properties of the critical points and sketch the phase portrait, including the special contours for which $K(x, y)=\frac{1}{4}$. Comment on the asymptotic behaviour, as $t \rightarrow \infty$, of solution trajectories that pass near each critical point, indicating whether or not any such solution trajectories approach from, or recede to, infinity.

Briefly discuss how the picture changes when $\alpha$ is made small and positive, using your result for $d K / d t$ to describe, in qualitative terms, how solution trajectories cross $K$-contours.

*Typos? Please submit corrections to this page on GitHub.*