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Numerical errors in a numerical solution of coupled ODEs

NOTE: This may be more of a "non-linear dynamics" problem than an ODE one.

Here goes...
I've recently been toying around with this system:

x' = y*e^{-(x^2+y^2)}
y' = -x*e^{-(x^2+y^2)} // (where "e^" denotes the exponential function)

I've noticed strange behavior that I can't seem to explain. I used a program I found online (PPlane) to graph this system. It _appears_ to consist of a number of closed circular orbits around the origin, as well as isolated fixed points (lying outside the closed orbit with the largest radius).
What I _think_ I know is that this system is reversible with a linear center (used the Jacobian), so we should theoretically expect these closed orbits, right? At least sufficiently close to the origin? Or..?
Instead of limit cycles, am I maybe actually seeing a continuous band of closed orbits? Or an algebraically decaying spiral? If so, how would I go about showing that?
It would also be worthwhile to know why I'm getting these isolated fixed points in PPlane. I don't think they should be there at all! Can you help me understand what is going on?
Oh I've almost forgotten.. I tried to rewrite the system in polar coordinates, but ended up with r' = 0 and theta' = -e^(-r^2). What gives? Eek!

Thanks for your time!

Solution Preview

Your solution you almost forgot to mention is perfectly correct.

As you have derived it yourself I suppose I am not to repeat your derivation.
It looks rather that the program you employed is not perfect.

I think it most likely that the problems of the program stem from the exponential factor e^(-r^2) which becomes very small very fast with r growing away from the center.
I do not have this program, but in any case its algorithm must be numerical rather ...

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