For a linear time-invariant system $\dot x = Ax,$ the inverse Lyapunov theorem asserts that if the origin is asymptotically stable, then a Lyapunov function in the form $V(x) = x^\top P x$ for some positive definite function $P.$
Is there a similar result for nonlinear systems? Namely, for a nonlinear dynamical system in the form $\dot x = f(x)$ such that the origin $x=0$ is globally asymptotically stable, is the existence of a Lyapunov function guaranteed? (Of course, even if the Lyapunov function exists, it might be very tricky to find.) If the existence of a Lyapunov function is not guaranteed, then are there any known counter-examples?