Periodic solution

Nov 2009
169
2
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?
 
Oct 2007
722
168
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?
Quick summarize...

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

=> \(\displaystyle 2rr' = 2x_1x_1' + 2x_2x_2'\) then sub in \(\displaystyle x_1'\) and \(\displaystyle x_2'\). 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.