# Poincaré-Bendixson theorem, periodic solutions/periodic orbits

According to my book (Hsu: ODE), a solution $\phi(t)$ to the system $x' = f(x)$ that is bounded for all $t \geq 0$ satisfies one of:

1) $\omega(\phi)$ contains an equilibrium, or

2) either $\phi(t)$ is periodic or $\omega(\phi)$ is a periodic orbit.

My question is: what is the difference between the solution being periodic and the limit set $\omega(\phi)$ being a periodic orbit?

• To me seems consistent. As a limit cycle has a periodic solution (once it attains the limit cycle) and the set of points in the phase space maps a closed orbit. (At least from my physicists point of view). – Chinny84 Jul 29 '15 at 10:48
• A solution spiralling in towards a limit cycle is an example of a non-periodic solution whose omega-limit set (i.e., the limit cycle) is a periodic solution. – Hans Lundmark Jul 29 '15 at 13:46
• Imagine solutions in polar coordinates $(r(t),\theta(t))$ with $\theta(t)=t$ and $r(t)=1-(1-r_0)e^{-t}$, then, for $r_0\ne1$, the solution never meets the unit circle, is not periodic, and has the unit circle as limit set, and the limit set $r=1$ is indeed a periodic orbit. – Did Jul 29 '15 at 13:49
• Oh so an example of a limit set as a periodic orbit is basically a function who tends to a periodic function in the limit? – Benjamin Lindqvist Jul 29 '15 at 17:46

As explained in comments, a non-periodic solution whose limit set is a periodic orbit is something spiraling toward a periodic orbit (either from the inside or from outside). An example of such a system is $$\begin{split} x' &= x\cos(x^2+y^2) - y \\ y' &= y\cos(x^2+y^2)+x\end{split}$$ The plot below (courtesy of Wolfram Alpha) shows two periodic orbits, one of which is attracting (and is the limit set for many non-periodic solutions) and the other one is repelling. 