Consider a dynamical system given by its [flow](http://en.wikipedia.org/wiki/Dynamical_system_%28definition%29) $\phi(t,x)$, where $t \in R$, $x \in R^n$ and $\phi: R \times R^n \to R^n$ is (say) differentiable. 


The *$\omega$-limit set*, $\omega(p)$, of a point $p \in R^n$ is the set of all $q \in R^n$ such that there exists a sequence $t\_0,t\_1,\dots$ with $t\_n \to \infty$ and 
$$\lim\_{n \to \infty} \phi(t\_n, p) = q.$$


The *$\omega$-limit set of a set* $X$ is simply the union of the $\omega$-limit sets of the points in $X$. 


It is not too hard to show that for any $X$, $\omega(\omega(X)) \subseteq \omega(X)$. 


However in all examples that I can think of, it always holds that $\omega(\omega(X)) = \omega(X)$. Is it possible to prove that, or is it false?


I guess either a proof or a counterexample should be already known, but I can't locate either. 


(Note: I already asked [a related question](https://mathoverflow.net/questions/47153/when-is-convergence-transitive) here on MO, but the question above is more specific and not answered there.) 



Omega-limit set of the omega-limit set
Stable homoclinic (or heteroclinic) loops in the plane provide a simple counterexample.


