  In this paper an operable, universal and simple theory on the attractiveness
of the invariant manifolds is first obtained. It is motivated by the Lyapunov
direct method. It means that for any point $\overrightarrow{x}$ in the
invariant manifold $M$, $n(\overrightarrow{x})$ is the normal passing by
$\overrightarrow{x}$, and $\forall \overrightarrow{x^{'}} \in
n(\overrightarrow{x})$, if the tangent $f(\overrightarrow{x^{'}})$ of the
orbits of the dynamical system intersects at obtuse (sharp) angle with the
normal $n(\overrightarrow{x})$, or the inner product of the normal vector
$\overrightarrow{n}(\overrightarrow{x})$ and tangent vector
$\overrightarrow{f}(\overrightarrow{x^{'}})$ is negative (positive), i.e.,
$\overrightarrow{f}(\overrightarrow{x^{'}}).
\overrightarrow{n}(\overrightarrow{x}) < (>)0$, then the invariant manifold $M$
is attractive (repulsive). Some illustrative examples of the invariant
manifolds, such as equilibria, periodic solution, stable and unstable
manifolds, other invariant manifold are presented to support our result.
