Definition of an $\omega$-limit point:
Let $\phi_t(p)$ be the orbit of the solution of the ODE $\dot{x}=f(x)$ which passes through the point $p$. We know that a point $x$ in $\mathbb{ R}^n$ is called an $\omega$-limit point of the orbit through the point $p$ if there is a sequence of numbers $t_1 \le t_2 \le t_3 \le · · ·$ such that $\lim_{i \to \infty } t_i = \infty $ and $\lim_{i\to \infty} \phi_{t_i}(p) = x$. From this definition, first and foremost, we understand that an $\omega$-limit point is "the limit point of a subsequence" constructed on the orbit. Obviously, this limit point may not be on the same orbit. I appreciate it if someone can answer the following questions:
Q1- Is that possible that an $\omega$-limit point is not on any orbit (i.e., it is not in the domain of definition of the solution (flow))? I guess that if the solution is forward complete (i.e., defined for all time $t\in(0,+\infty)$), then all the omega limit points are always in the domain of definition of the solution.
Q2- How should we define the $\omega$-limit point for the case when $t$ cannot go to $\infty$, i.e., the maximal interval of existence of the solution (i.e., domain of $t$) is $(-\infty, a)$ where $a<\infty$?
Q1. It is quite easy to construct examples where $f$ is undefined at an $\omega$-limit point of some orbit (which is not on the orbit itself). A trivial way to do so is just to remove that point from the domain of $f$. There are also less trivial examples where $f$ does not have a limit as you approach the point in question, so it is impossible to redefine $f$ at the $\omega$-limit point to make it continuous there.
Q2. By definition, an orbit that is not defined for $t \to +\infty$ can't have an $\omega$-limit point.