Let $x:\mathbb{R}^+\to \mathbb{R}^n$ such that $\lim_{t\to +\infty} x(t)=0$. Under which conditions can I say that there exists a Lipschitz-Continuous function $f$ such that $\frac{d}{dt}x(t)=f(x)$ and such that $f$ has a unique equilibrium point in $x=0$?
In other words, given a function $x(t)$ (or a sequence of functions $x_n(t)$) how can I prove that it is (they are) generated by an ODE of the form $\dot x = f(x)$?