For smooth continuous dynamical system,
$$\dot{x} = f(x),$$
on manifold $\mathcal{M}$, can I say it is partitioned by countably many basins of attraction?
Motivation
I want to prove something which requires the following conjecture
- the manifold associated with the dynamical system can be partitioned by basin of attraction, which constitutes the manifold $\mathcal{M}$ almost everywhere, say if we assume a Lebesgue measure.