I'm probably missing something stupid here, regarding periodic solutions to ODE's.
The ode is $$\dot{x}=-\nabla f$$
I can see that a solution will require $\frac{df}{dt}=-\|\nabla f\|^2$. Buy why does this prevent existence of such solutions?
More generally, why are we interested in the behavior of $f$ rather than $x$?
For any solution you get $$ \frac{d}{dt}f(x(t))=f'(x(t))\dot x(t)=-f'(x(t))∇f(x(t))=-\|∇f(x(t))\|^2 $$ so that either the solution is stationary or decreases the level of $f$, preventing the return to a previous point.