Given $C^1-$function $f:\mathbb{R}^2\to \mathbb{R}^2$, consider differential equation $\dot{x}=f(x)$, which may have finite singularities $x_i, 1\leq i \leq n$. Let $\varphi^t(x)$ denotes the corresponding solution of the equation, which is also called the flow of the dynamical system. We know that a point $y$ is a $\omega-$limit point of x for $\varphi^t$ if there exists a sequence $t_k$ going to infinity, s.t. $lim_{k\to \infty}d(\varphi^t(x),y)=0$. The set consists of all of these points is called the $\omega-$limit set of $x$, and denoted $\omega(x)$. The $\alpha-$ limit set is defined similarly, only with $t_k$ goes to minus infinity.
For a point $y\in \mathbb{R}^2$ with $f(y)\neq 0$, which is not a period point, it's asked to prove that either $\omega(y)\cap \alpha(y)=\emptyset$ or $\omega(y)=\alpha(y)=\{x_i\}$, the finite points set.
We know that $\omega(y)=\cap_{T\geq0}\overline{\cup_{t\geq T}\{\varphi^t(y)}\}, \alpha(y)=\cap_{T\leq 0}\overline{\cup_{t\leq T}\{\varphi^t(y)}\}$. Thus if their intersection is nonempty, then $\exists$ {$t_k$} goes to infinity and {$s_k$} goes to minus infinity, s.t. $lim_{k\to\infty}d(\varphi^{t_k}(y)-\varphi^{s_k}(y))=0$. But then I got stuck and cannot go further. Hope someone could help. Thanks!
