While I understand that doing the above is not possible *in general*, I would like to know more about how to proceed when it *is* possible. That is, what are the 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. Thank you very much in advance to anyone who took the time to read this.


**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."$^2$.


P.S. I had originally asked this on [Mathermatics StackExchange](https://math.stackexchange.com/q/4497042/798206), but a kind user recommended that it would be better for me to ask this here!


  

  



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**Footnotes:**  



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 $[a,b]^n$.
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).



Analytically characterizing basins of attraction boundaries and sizes
Despite having searched for such a reference for quite a while before asking my question, I ironically found a reference that seems to fit the bill! [Alberto and Chiang (2015, p. 198)](https://www.cambridge.org/core/books/stability-regions-of-nonlinear-dynamical-systems/522A4FACB373591173A78C710BF004D8) looks like a great resource for characterizing/analyzing basins of attraction and their boundaries, sizes, etc. The two authors seem to also have other related papers focusing on special cases (e.g. their [2011 paper](https://ieeexplore.ieee.org/document/6070954?reload=true) focuses on general autonomous nonlinear dynamical systems)


These resources seem great, but please do let me know if you know of others! In the meanwhile, I hope this helps any passers by :)


