Differential Equations | Part IA, 2003

For all solutions of

x˙=12αx+y2y3y˙=x\begin{aligned} &\dot{x}=\frac{1}{2} \alpha x+y-2 y^{3} \\ &\dot{y}=-x \end{aligned}

show that dK/dt=αx2d K / d t=\alpha x^{2} where

K=K(x,y)=x2+y2y4K=K(x, y)=x^{2}+y^{2}-y^{4}

In the case α=0\alpha=0, analyse the properties of the critical points and sketch the phase portrait, including the special contours for which K(x,y)=14K(x, y)=\frac{1}{4}. Comment on the asymptotic behaviour, as tt \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 α\alpha is made small and positive, using your result for dK/dtd K / d t to describe, in qualitative terms, how solution trajectories cross KK-contours.

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