I'm studying a nonlinear autonomous ODE system in several variables, $\dot{x}=f(x)$ where $f$ is a Lipschitz vector field. For the unique solution $x(t)$ that starts at $x_0$, I'd just like to say that $\lim_{t\to\infty} x(t)$ exists.
To do this, the only general method that I know is to find a Lyapunov function, or, in alternative, one can "work it around explicitly" and take the limit. Unfortunately, both approaches seem undoable in my case. So my question is: which other general methods/theorems can I use to prove existence of the limit above?
So you want to know if the solution to $\dot{x}=f(x)$ can be extended to infinity. One sufficient condition is that there exists $C>0$ s.t. $$\langle x,f(x)\rangle\leq C(1+\|x\|^2) \quad \forall x.$$ A classical example of the DE that violates this condition is $\dot{x}=x^2$