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.
enter image description here


2 Answers 2


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 $x \times y \times t$.

Curve apparent crossing


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:

Non-autonomous system

And for an autonomous system: enter image description here

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."

  • 1
    $\begingroup$ Hi Aditya, and welcome to the Physics SE! The equations become much easier to read, search and edit when mathjax is used. It'd be great if you could use it in your next posts. $\endgroup$
    – stafusa
    Jan 21, 2021 at 10:25
  • $\begingroup$ @stafusa Thanks for the suggestion $\endgroup$ Jan 22, 2021 at 5:03

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