Consider the system $\frac{dx}{dt} = y$, $\frac{dy}{dt} = -x^2y - x^3$ which has $(0,0)$ as a (unique) critical point. I aim to show that $(0,0)$ is stable (at least that's what Wolfram Alpha sketches show), however linearized analysis does not work here since $0$ is a double eigenvalue. So I tried to find a Lyapunov function (e.g. the form $cx^4 + dy^2$) which would work but did not succeed. Any idea on how to do this?
Any help appreciated!