Sketch a phase portrait of these of the dynamical system in Cartesian. Am I missing a trick?

by Gragbow   Last Updated February 19, 2018 19:20 PM

I am asked to sketch the phase portrait of the system

$\dot r =r(1-r^2) \\ \dot \theta=r^2((1-r^2)^2+sin^2(\theta))$

but in Cartesian.

I am wondering whether I have to actually convert these coordinates into Cartesian before plotting, or if there is some trick that I am missing.

I only ask this because it's taking a lifetime to convert these coordinates, and I don't even think I'm getting them right.

I get

$\dot x = \frac{x^2+y^2}{(x+y)}\Big(x(1-(x^2+y^2)\Big)-y\Big((1-(x^2+y^2))^2+\frac{y^2}{x^2+y^2}\Big)$

which just doesn't seem right to me

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