Definition:
The $\omega$-limit set $L_{\omega}\left ( x \right )$ of $x \in \mathbb{M}$ >is the set of $y \in \mathbb{M}$ which for each y there exists a strictly increasing unbounded sequence of times
$t_{0}<t_{1}<t_{2}<\cdot \cdot \cdot $
such that $\lim_{n\rightarrow \infty}\left \| \Psi_{t_{n}}\left ( x \right )-y \right \|=0$
Definition:
Let $S\subseteq \mathbb{M}$ be an invariant set.
$W^{s}_{s}\left ( S \right )=\left \{ y \in\mathbb{M}:L_{\omega}\left ( y \right )\subseteq S \right \}$
-inset, stable set, stable manifold
and
$W^{u}\left ( S \right )=\left \{ y \in \mathbb{M}:L_{\alpha}\left ( y \right )\subseteq S \right \}$
-outset, unstable set, unstable manifold
I would like to request for a more contextual explanation of the above definition. Would someone kindly do so? Geometrical intuition would be even helpful. Thanks in advance.
That is an unusual way of describing stable/unstable manifolds, and actually wrong in general.
How about say $x'=-x^2$ that has no stable manifold but that according to this definition has? Simply it is not standard (nor correct).
On the other hand, the intuition is clear: stable manifolds of $S$ should more or less be (although in general are not) the set of points that converge to the set $S$.