Use the PoincarBendixson theorem to show that the plane autonomous system x Px y x(x2 y2
Chapter 11, Problem 20(choose chapter or problem)
Use the Poincaré-Bendixson theorem to show that the plane autonomous system
\(\begin{aligned}
&x^{\prime}=\epsilon x+y-x\left(x^{2}+y^{2}\right) \\
&y^{\prime}=-x+\epsilon y-y\left(x^{2}+y^{2}\right)
\end{aligned}\)
has at least one periodic solution when \(\epsilon>0\). What occurs when \(\epsilon<0\)?
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