I have some doubts about Liouville theorem, probably its just something conceptual.
So: I know that for a system in which Liouville’s theorem holds, the volume in the phase space is conserved.
But the conservation of the volume does immediately imply the absence of asymptotically stable points.
However, if the Hamiltonian is time dependent and in particular of its time derivative is negative along the phase curves, a system does possess asymptotically stable points.
For that system, Hamilton equations still hold, hence it’s an Hamiltonian system.
But Liouville’s theorem doesn’t hold anymore.
My question: for what kind of systems does Liouville’s theorem hold? For example: the dumped oscillator has one asymptotically stable point.