In Salamon's Connected Simple Systems, p.8, the author writes that the $\omega$-limit of a set $Y$ inside a flow $\Gamma$ has the two equivalent descriptions $$ \omega(Y) = I(\overline{Y \cdot [0,\infty)}) = \bigcap_{t > 0} \overline{Y \cdot [t,\infty)}.$$ The notation $I(X)$ means the maximal invariant subset inside $X,$ i.e., $I(X) = \{\gamma \in \Gamma : \gamma \cdot \mathbb{R} \subset X \},$ i.e. the union of all the flow curves that lie entirely inside $X.$
However it seems to me that these two descriptions don't agree. Consider the following flow on $\mathbb{R}^2$ (sorry for the horrible drawing):
This is just the flow of $\partial/\partial x$ modified inside a small ball so as to possess a rest point at $(x_0,0)$ for some $x_0 > 0.$ (The rest point is the black dot.) Consider, say, the set $Y = \{0\} \times [-1,1].$
On one hand we have $Y \cdot [0,\infty) = \left([0,\infty) \times [-1,1]\right) \setminus \left((x_0,\infty) \times \{0\}\right),$ so that its closure is the whole band $[0,\infty) \times [-1,1]$ and $\omega(Y) = I(\text{band}) = [x_0,\infty) \times \{0\}.$
On the other hand, if we take any $t > x_0,$ then the set $\overline{Y \cdot [t,\infty)} = \left([t,\infty) \times [-1,1]\right) \cup \{(x_0,0)\},$ so that the intersection $\bigcap_{t > 0} \overline{Y \cdot [t,\infty)}$ has no chance of being equal to $\omega(Y).$
What am I missing here?
I think that the only effect that you haven't taken into account is that all trajectories near the equilibrium experience slowdown and as a result the set $Y \cdot \lbrack t, +\infty ) $ (at least its left border, $Y \cdot \lbrace t \rbrace$) shaped a little differently than a union of a stripe with a point. I hope that this picture will help illustrate my idea:
And because every $Y \cdot [t, +\infty )$ is shaped like this from some moment, the $\bigcap_{t > 0} \overline{Y \cdot [t,\infty)}$ gives the same $\omega(Y) = [x_0,\infty) \times \{0\}$.