# Does Asymptotic Stability Imply the Existence of a Lyapunov Function for a Nonlinear System?

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?

Let $$x=0$$ be an asymptotically stable equilibrium of $$\dot{x}=f(x)$$. Then there is a smooth positive definite function $$V(x)$$ and a continuous positive definite function $$W(x)$$ such that: $$\frac{\partial V}{\partial x}\,f(x)=-W(x)$$