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

This is a nice example of what a nonlinear term can do to a stable, but not asymptotically stable, equilibrium. It helps to introduce the polar radius $\rho=\sqrt{y_1^2+y_2^2}$, because this function satisfies the ODE 
$$\frac{d\rho }{dt} = \frac{y_1}{\rho}\frac{dy_1}{dt}+\frac{y_2}{\rho}\frac{dy_2}{dt} = \rho^3\sin \frac{\pi}{\rho} \tag1$$
The analysis of equilibria of this ODE tells you  about the orbits of the original system: 


*

*There are stable closed orbits of radius $\rho=\dfrac{1}{2k }$, $k=1,2,\dots$ 

*There are unstable closed orbits of radius $\rho=\dfrac{1}{2k-1 }$, $k=1,2,\dots$ 

*In between, the orbits are spirals converging to the nearest stable closed orbit. 

*Outside, in the region $\rho>1$, the orbits go off into infinity in a hurry (in such a hurry that they get there in finite time).


The plot given by Amzoti  illustrates all of the above points.
