Differential Equations | Part IA, 2002

For any solution of the equations

x˙=αxy+y3(α constant )y˙=x\begin{aligned} &\dot{x}=\alpha x-y+y^{3} \quad(\alpha \text { constant }) \\ &\dot{y}=-x \end{aligned}

show that

ddt(x2y2+12y4)=2αx2.\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 tt \rightarrow \infty, in the case α=0\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 α=0\alpha=0, (b) in the case α=0.1\alpha=0.1, and (c) in the case α=0.1\alpha=-0.1.

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