I understand that doing the above is not possible in general. However, when it is possible, what are common methods people use to analytically characterize basin of attraction boundaries (i.e. via a closed-form expression) or their (Lebesgue) measures? Regarding the latter: I do not necessarily need an exact expression for their Lebesgue measure; anything that lets me say something analytically about the "size" of a basin of attraction would be great!) Any references or guidance would be immensely appreciated.
More Details: I have a nonlinear, first-order, autonomous system of $n\in\{2,3,...\}$ ODE defined on $[a,b]^n$, where $-\infty<a<b<\infty$.$^1$ I currently cannot solve for the exact solution, but I have identified the set of locally asymptotically stable (LAS) steady states $\{\vec{s}_1,...,\vec{s}_k\}$, where $k<\infty$. Each LAS steady state's basin of attraction seems to be very "well-behaved."$^2$.
Footnotes:
$\quad$ 1. That is, this system is only defined at points in $[a,b]^n$, and any trajectory originating in $[a,b]^n$ (and obeying this system) always remains in $[a,b]^n$.
$\quad$ 2. They are all simply connected. $k-1$ basins are convex sets; the remaining basin -- while non-convex -- just consists of the remaining space. The boundaries are "simple looking" and "smooth" (both in the colloquial sense). Note that these boundaries are not "flat" in the sense that they cannot be expressed as a hyperplane (or a finite union thereof).