This isn't really research-level and you may have better luck on math.stackexchange.com (as per the FAQ here), but FWIW you may perhaps gain some insight by sonsidering the example $f\colon \mathbb{R}^2\to\mathbb{R}^2$ given by $f(x,y) = (2x, y/2)$, and using the definitions to work out the omega limit set and stable manifold of $(0,0)$. Then try $(0,1)$ and $(1,0)$.
– Vaughn ClimenhagaAug 29 '12 at 15:40

As @Vaughn Climenhaga said, this question is not research level. But maybe a quick answer is that if $x$ is in the omega limit set of $p$, then $p$ is in the stable manifold of $x$ and vice versa.
– user7807Aug 31 '12 at 22:40