How to calculate $\omega$-limits I'm trying to learn how to determine the omega limit $\omega(p)$ of a set, in order to do this I need help in a simple example. Let $p\in \mathbb R^2$ in the case of the field $Y=(Y_1, Y_2)$, given by:
$Y_1=-y_2+y_1(y_1^2+y_2^2)\sin\left(\dfrac{\pi}{\sqrt{y_1^2+y_2^2}}\right)$
$Y_2=y_1+y_2(y_1^2+y_2^2)\sin\left(\dfrac{\pi}{\sqrt{y_1^2+y_2^2}}\right)$
By definition Let $\Delta$ be an open subset of the euclidian space $\mathbb R^n$ and $X:\Delta\to \mathbb R^n$ a vectorial field of $C^k$ class, $k\geq 1, $ and $\varphi(t)=\varphi(t,p)$ an integral curve of $X$ passing by $p$ defined in its maximal interval. 
$w(p)=\{ q\in \Delta;\exists(t_n)$ with $t_n\to \infty$ and $\varphi(t_n)\to q,$ when $n\to\infty\}$.
I need help, I can't use this definition to solve this question, it seems to me really complicated.
Thanks so much.
 A: The following kind of analysis may help you to answer this question. First, some observations for system. You have only one steady state, that is located at $(0, 0)$. This vector field is a perturbation of vector field $Y_c = (-y_2, y_1)$, which integral curves are the family of concentric circles around the origin. You can write exact solution for $Y_c$. Also you can see, that the perturbation vanishes at some of circles $y_1^2 + y_2^2 = const$; you can directly check, that solution for vector field $Y_c$ at these circles is also an integral curve for a v.f. $Y$. So, you have a countable set of closed trajectories around the origin. Second, you can go to polar coordinates $(\rho, \varphi)$ and/or look what happens with $\frac{d\rho^2}{dt} = \frac{d}{dt}(y_1^2+y_2^2)$. Guts feeling tells me that it'll be independent of $\varphi$ variable and after that all you have to do is look at one-dimensional dynamics. From this you'll get an info about limit cycles and their stability. That'll be enough to determine the $\alpha$- and $\omega$-set for any trajectory at plane. 
