Consider the system

$$\begin{aligned} &\dot{x}=y+a x+b x^{3} \\ &\dot{y}=-x \end{aligned}$$

What is the Poincaré index of the single fixed point? If there is a closed orbit, why must it enclose the origin?

By writing $\dot{x}=\partial H / \partial y+g(x)$ and $\dot{y}=-\partial H / \partial x$ for suitable functions $H(x, y)$ and $g(x)$, show that if there is a closed orbit $\mathcal{C}$ then

$$\oint_{\mathcal{C}}\left(a x+b x^{3}\right) x d t=0$$

Deduce that there is no closed orbit when $a b>0$.

If $a b<0$ and $a$ and $b$ are both $O(\epsilon)$, where $\epsilon$ is a small parameter, then there is a single closed orbit that is to within $O(\epsilon)$ a circle of radius $R$ centred on the origin. Deduce a relation between $a, b$ and $R$.