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Show that the system

x'= x-y-x[x^2 + (3/2)y^2]

y'= x+y -y[x^2 + (1/2)y^2]

has at least one periodic orbit.

I know that I need to apply Poincare-Bendixson Theorem. I can prove the first three points of it easily, but to create a trapping region, I believe that I need to switch this system to polar. I know that I need to make the substitutions y=r*sin(θ), dy= sin(θ)*dr + r*cos(θ) θ', x= r*cos(θ), dx= cos(θ)*dr - r*sin(θ)θ'. But when I do make the substitution, it makes the equations worse than before. Is there some other way to come up with a r min and r max for the trapping region? I just assumed it was a polar type question because its phase plane has a limit cycle on it.

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# Change ODE system to Polar to apply Poincare-Bendixson

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