I am trying to find Lyapunov function for $$\begin{cases}\dot{t} = y\\\dot{y} = t^2-t\end{cases}$$ I tried common examples but, maybe I was wrong in my computations, couldn't derive anything.
Could you help? Is there any approaches that cover some easy situations like this one (when variables are "separated")?
Please allow me to write $(x,y)$ instead of $(t,y)$.
Any system of the form $$ \dot x = y ,\quad \dot y = -V'(x) $$ (which is equivalent to the Newton-type second-order ODE $\ddot x = -V'(x)$ where $V(x)$ is the potential energy) is a Hamiltonian system generated by $H(x,y) = \tfrac12 y^2 + V(x)$, so $H$ is a constant of motion (and thus also a weak Lyapunov function in some neighbourhood of the origin, if it happens to have a local minimum at the origin, like in this case where $H(x,y) = \tfrac12 y^2 + \tfrac12 x^2 - \tfrac13 x^3$).