Paper 2, Section II, C

Consider the nonlinear system

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

(a) Show that $H=H(x, y)=x^{2}+y^{2}-y^{4}$ is a constant of the motion.

(b) Find all the critical points of the system and analyse their stability. Sketch the phase portrait including the special contours with value $H(x, y)=\frac{1}{4}$.

(c) Find an explicit expression for $y=y(t)$ in the solution which satisfies $(x, y)=\left(\frac{1}{2}, 0\right)$ at $t=0$. At what time does it reach the point $(x, y)=\left(\frac{1}{4},-\frac{1}{2}\right) ?$

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