how to draw graphs of ODE's In order to solve this question How to calculate $\omega$-limits I'm trying to learn how to draw graphs of ODE's. For example, let $p\in \mathbb R^2$ in the case of the field $Y=(Y_1, Y_2)$, given by:
$Y_1=-y_2+y_1(y_1^2+y_2^2)\sin\left(\dfrac{\pi}{\sqrt{y_1^2+y_2^2}}\right)$
$Y_2=y_1+y_2(y_1^2+y_2^2)\sin\left(\dfrac{\pi}{\sqrt{y_1^2+y_2^2}}\right)$
I need help.
Thanks so much

( Explanation of yesterday's post implication reg constant of integration)
These are Limit Cycles in Control theory. There are three limiting (asymptotic) loci as sketched by Amzoti. However, caution is needed in choosing boundary conditions if the integrate is desired to be in a neat, closed or elegant form involving circular trig functions. Else, the graphs would look displaced from what might be considered their "natural" position. I mention this as OP asked how to draw the integrands or phase portraits.  
(I am using $ y_1 = x$, and $y_2 = y$).
if x(0) is given, then better not to choose y(0) arbitrarily because: 
$(y'-y) = (x'-x) $ and $(x'^2 + y'^2) = (x^2 + y^2) (1+R^2)$,
where $R = (x^2 + y^2) \sin (\pi/ \sqrt{x^2 + y^2})$
together define functional dependance between x and y. ( The f(x,y)= 0 result can be posted here later on by the OP to gain further clarity). Constant of integration is to be chosen for the most "natural"  or elegant form. In general of course, the constants can be chosen quite arbitrarily.
I give a simpler example of what I mean.
Take $y' = \sin (x)/m$ and $x'= \sin (y)$. Integrating, $\cos (x) = m \cos (y) +$ arbitrary constant,  which should be set to zero if the phase portraits do not get an asymmetrical look displaced away from argument values $0, \pi/2, \pi, 2 \pi$ etc. periods. This may be "desirable" but not essential for generality. Here $\cos (x)/\cos (y)= m$ looks more elegant. The advantage becomes apparent further on when trying to form higher order decoupled equations  $F( x, x'' ) = 0$,  $G(y, y'' ) = 0$, towards a fuller or more comprehensive situation.
