As stated in the title I have the following system of ODE's
$$\begin{aligned}\dot{x} &= y,\\ \dot{y} &= -x +\frac{3}{2}x^2\end{aligned}$$
I have already found the stationary points $(0,0)$ and $(2/3,0)$ and I have the Lyapunov-function $L(x,y)=x^2-x^3+y^2$. I now want to prove that all solutions close to the origin are periodic and that there is a non-periodic, bounded solution. I think I got the first claim, as the Lie-derivative of $L$ is $0$ everywhere, so we have a constant of motion.
However, I don't see how I would prove the second claim using $L$. It doesn't even seem logical to me, by looking at the phase diagram I would have guessed that all solutions are either unbounded or periodic around $(0,0)$ or stationary.
Any help is appreciated!