2.II.8D

For any solution of the equations

$\begin{aligned} &\dot{x}=\alpha x-y+y^{3} \quad(\alpha \text { constant }) \\ &\dot{y}=-x \end{aligned}$

show that

$\frac{d}{d t}\left(x^{2}-y^{2}+\frac{1}{2} y^{4}\right)=2 \alpha x^{2} .$

What does this imply about the behaviour of phase-plane trajectories at large distances from the origin as $t \rightarrow \infty$, in the case $\alpha=0$ ? Give brief reasoning but do not try to find explicit solutions.

Analyse the properties of the critical points and sketch the phase portrait (a) in the case $\alpha=0$, (b) in the case $\alpha=0.1$, and (c) in the case $\alpha=-0.1$.

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