1. Periodic solution

Use the Poincaré-Bendixon theorem to show that the following differential equations
$x_1' = 2x_1 - 2x_2 -x_1(x_1^2+x_2^2)$
$x_2' = 2x_1 + 2x_2 -x_2(x_1^2+x_2^2)$

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?

2. Originally Posted by EinStone
Use the Poincaré-Bendixon theorem to show that the following differential equations
$x_1' = 2x_1 - 2x_2 -x_1(x_1^2+x_2^2)$
$x_2' = 2x_1 + 2x_2 -x_2(x_1^2+x_2^2)$

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?
Quick summarize...

You first have to show that there is a trapping region. Do this by setting $r^2 = x_1^2 + x_2^2$.

=> $2rr' = 2x_1x_1' + 2x_2x_2'$ then sub in $x_1'$ and $x_2'$. Show that a disk of radius r is a trapping region by showing that r' <= 0...

Then the theorem says if there are no critical points within this trapping region (or if there is either an unstable spiral or unstable node) there will be a limit cycle (periodic solution). So just show one of those is true.