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I am trying to understand why the (positive or omega) limit set of a non-autonomous dynamical system \begin{equation} \dot{x}=f(t,x) \label{ftx} \tag{1} \end{equation} is not necessarily (positively) invariant. Assume that $f$ is piecewise continuous in $t$ for each fixed $x$, and that $f$ is locally Lipschitz in $x$ uniformly over $t$.

I have seen the proof for the autonomous case in this question, and I imagine that, for the non-autonomous case, a set $\bar{\omega}$ may belong to the positive limit set of \ref{ftx} for a given initial time $\bar{t}$, but if the initial time is changed to some $\hat{t}$, then it is possible that the solution will enter $\bar{\omega}$ and leave it.

Am I correct? Or is there something I'm missing that will break in the aforementioned proof?

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    $\begingroup$ Take $$ \dot{x} = \frac{1}{1 + t^2}.$$ What are "$\omega$-limit sets" in this case? Are they invariant? $\endgroup$ – user539887 Apr 4 '18 at 7:36
  • $\begingroup$ @user539887 for the example you give, the $\lim_{t\to \infty}\phi(t,t_0,x_0)=\pi/2+x_0-\arctan(t_0)$, right? So your point is that the limit sets may depend on $t_0$ and, consequently, may not be invariant? $\endgroup$ – xuva Mar 24 at 17:44
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    $\begingroup$ Yes, that was my intention. $\endgroup$ – user539887 Mar 25 at 19:11

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