Does the solution for bounded non autonomous system converge?

Question

I wonder if the solution of scalar non-autonomous system

$\dot x(t)=F(x,t),\quad x(0)=x_0$

has a limit if $x(t)$ is uniformly bounded in time $t$.

Here $F$ is a Lipschitz function.

Can we say $x(t)$ is monotone after some point $t_*$?

Answer 1

Take the system $\dot{x}(t) = \sin t$. The solution $x(t) = x(0) - (\cos t -1) $ is bounded, but $x$ is not monotonic.