Can phase trajectories intersect for non-autonomous system?

Question

There has been enough discussion about intersection of phase trajectories in autonomous system,where the system wasn't time dependent. And we came to the conclusion that, at a point in space there can't be two futures and two stories of history over the course of evolution, hence phase trajectories can't intersect. But I haven't found any conclusive discussion of the same about non autonomous system(time dependent). Though in some research work, I found the following statement:

"non-autonomous system trajectories can have self intersection and two different trajectories can intersect in later time."

Please explain the reason for the same. I've attached a screenshot from that paper.

Answer 1

Yes, trajectories of non-autonomous systems can cross in phase space.

One way to understand why is to consider that the system's explicit time dependence means that the time $t$ is needed to unambiguously determine the state of the system, i.e., the "complete" phase space includes an extra dimension $t$ besides, say, $(x,y)$: so what appears as a crossing in space $x \times y$ only seems so because it's a projection (over $x \times y$) of the (non-self-crossing) trajectory in the 3-D space $x \times y \times t$.

Answer 2

As the research paper rightly points that for non-autonomous system trajectories may intersect. The gist of this concept lies in the definition of autonomous and non-autonomous system. For a non-autonomous system:

And for an autonomous system:

Thus for a non-autonomous system, the equation governing the evolution of system changes with time due to the variable '$t$' in the equations. However, this doesn't happen for an autonomous system. Thus an argument similar to your explanation can be raised i.e.:

"As the rules governing the evolution of system with time, change with time so we still have a unique past and future even if these trajectories intersect."