Use the Poincaré-Bendixon theorem to show that the following differential equations

$\displaystyle x_1' = 2x_1 - 2x_2 -x_1(x_1^2+x_2^2)$

$\displaystyle x_2' = 2x_1 + 2x_2 -x_2(x_1^2+x_2^2)$

have a nontrivial periodic solution.

Actually I would liek to solve the problem myself, but I don't really understand how this works. Anyone has any hints? And can someone summarize this theorem in reall easy words?