On the first Lyapunov method, when the linearization fails

Question

I have been trying to apply the first Lyapunov method to decide about the stability of the origin for the following system \begin{equation*} \dot{x}=\sqrt[3]{-x}. \end{equation*}

However, the linearization does not exist around the origin, since \begin{equation*} \dfrac{d \sqrt[3]{-x}}{d x}=-\dfrac{1}{3\sqrt[3]{x^2} }. \end{equation*}

What can I conclude?

Answer 1

Try the 2nd method. Start by guessing a Lyapunov function candidate, something simple such as $V = x^2$ to start with.