Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by $$\dot{x}=F(x(t))$$
An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of $S$ is called a Morse decomposition of $S$ if there exists an increasing sequence $\emptyset=A_{0} \subset A_{1} \subset \cdots \subset A_{l}=S$ of attractors in $S$ called Morse filtration of $ \mathcal{M}$ , such that $$M_{k}=A_{k} \cap A_{k-1}^{*}, \quad 1 \leq k \leq l $$
Hence, we can generally define a pairwise disjoint decomposition of $S$ by $$S= \dot{\bigcup}_{i<j=1}^N M_i \dot{\cup} C(M_i,M_j)$$ where $C(M_i,M_j)$ denotes the connecting orbits between the morse sets.
Lastly, we define the basin of attraction of an attractor as the largest open subset of $S$ such that $\omega(U)=A$, where $\omega$ denotes the limiting set.
My question is the following: What is the representation of the basin of attraction of an attractor in terms of morse sets? Cann we define a similiar decomposition of $S$ (as above) in terms of basins of attraction?