Does a 2D stable limit cycle always contain an unstable equilibrium point?

I believe that the answer is yes (provided there are no further limit cycles within the first one).

Let there be a stable limit cycle with no other limit cycles within it. By reversing time, $t \rightarrow -t$, we get a new system with trajectories bounded to the interior of the limit cycle. Using, say, Brouwer’s fixed point theorem, one can show that there must be a fixed point within the region.

However, when reading different formulations of the Poincaré–Bendixson theorem, I often encountered the statements like this:

If the trajectories are bounded and there are no fixed points, then the trajectories must converge to a limit cycle.

Are these are just unlucky/incorrect formulations or there is something in it?

Well, if the trajectories are confined to some region which does contain a fixed point, then it may happen that they all converge to that fixed point, so that there are no limit cycles in the region. So by assuming that there are no fixed points, you rule out that possibility.

But note that when you apply Poincaré–Bendixson, the trapping region is usually an annulus (or something topologically equivalent to that), so you don't consider the whole region inside the potential limit cycle. As you say, there is necessarily a fixed point inside the inner circle of the annulus, but that point doesn't belong to the annulus.

• Dear, @Hans-Lundmark, sorry for my ignorance, but what would be the reason to exclude the inner part of the annulus? That is to say, why do we consider an annulus and not a circle? – Dmitry Dec 15 '17 at 9:54
• Simply to obtain a region where there are no fixed points. – Hans Lundmark Dec 15 '17 at 10:23

If the trajectories are bounded and there are no fixed points, then the trajectories must converge to a limit cycle.

Well, to what region does “there are” refer in such a statement? In all formulations I am aware of, this would refer to the limit set of the trajectories in question, and thus be oblivious to equilibria located within the limit cycle.

Let there be a stable limit cycle with no other limit cycles within it. By reversing time, $t \rightarrow -t$, we get a new system with trajectories bounded to the interior of the limit cycle. Using, say, Brouwer’s fixed point theorem, one can show that there must be a fixed point within the region.

Ironically, you can apply the Poincaré–Bendixson theorem (e.g., in Wikipedia’s formulation) to the interior of the now-unstable limit cycle to show that there has to be at least one stable or saddle equlibrium¹. Going back to the forward variant of the system, this becomes an unstable equilibrium.

Note that it can happen that the only unstable equilibria you have within your limit cycle are saddles. For example, you could have a saddle with two repelling homoclinic orbits, each of which encloses a stable equlibrium.

¹ If there were only fully unstable equilibria, you could perform reductio ad absurdum by excluding their neighbourhoods from the trapping region of the Poincaré–Bendixson theorem.