$\omega$ and $\alpha$ - limit sets I'm reading a book that defines the $\omega$ and $\alpha$ -  limit sets of a differential equation respectively as :
$\alpha (x)$ = $\cap_{y \in \gamma (x)}$ $\overline \gamma^{-} (y)$
$\omega (x)$ = $\cap_{y \in \gamma (x)}$ $\overline \gamma^{+} (y)$
with
$\gamma^{+} (y)$ = {$\varphi _t (x) $ : $t \in I_x \cap R^+$}
$\gamma^{-} (y)$ = {$\varphi _t (x) $ : $t \in I_x \cap R^-$}
$\varphi _t (x_0)$ being the solution x(t), where x(0)=$x_0$
What is the interpretation of this definition graphically? I normally use the following definition with limits: 
http://en.wikipedia.org/wiki/Limit_set
Can you please give me a guided example of how to apply the upper definition? How are they equivalent? I'm missing something.
Thank you.
 A: let $x_n $, $n\in\mathbb{N}$, be a sequence in $\mathbb{R^m}$ for some $m$. then we can consider the set of subsequences limit points, i.e. 
Set of subsequence limit points of $x_n =\lbrace x\vert\exists n_k\rightarrow\infty \text{ such that }   x_{n_k}\rightarrow x\rbrace$.
you can show that the abov set is closed. in fact we have:
Set of subsequence limit points of $x_n=\bigcap\limits_{n=1}^{\infty}\overline{\lbrace x_k\vert k\geq n\rbrace}$.
Now let $f:\mathbb{R^m}\rightarrow\mathbb{R^m}$ be a homeomorphism. then we can consider the forward orbit of a point x,i.e the sequence of forward iterates of $\lbrace f^n(x)\vert n\geq0\rbrace$ and the backward orbit of a point x,i.e $\lbrace f^n(x)\vert n\leq0\rbrace$.
Now for the both above sequences we define $\omega(x)$ and $\alpha(x)$ as the set of subsequences limit points of the forward and backward orbits of x. By the above argument we have:
$\omega(x)=\lbrace y\vert\exists n_k\rightarrow\infty \text{ such that }   f^{n_k}(x)\rightarrow y\rbrace=\bigcap\limits_{n=1}^{\infty}\overline{\lbrace f^k(x)\vert k\geq n\rbrace}$
and something similar for $\alpha(x)$.
Example:consider the following function:

two marked points are fixed and the arrows indicate the direction from $x$ to $f(x)$. It's clear that the set of periodic points=two marked points $(a, b).$
and its easy to see that $\omega(a)=\lbrace a\rbrace=\alpha(a)$ and $\omega(b)=\lbrace b\rbrace=\alpha(b)$.
